| L(s) = 1 | − 3-s + 5·4-s − 7-s − 5·12-s + 15·16-s − 17-s − 19-s + 21-s − 23-s + 5·25-s − 5·28-s − 37-s − 41-s − 47-s − 15·48-s + 51-s + 57-s + 35·64-s − 67-s − 5·68-s + 69-s − 5·75-s − 5·76-s − 79-s − 83-s + 5·84-s − 5·92-s + ⋯ |
| L(s) = 1 | − 3-s + 5·4-s − 7-s − 5·12-s + 15·16-s − 17-s − 19-s + 21-s − 23-s + 5·25-s − 5·28-s − 37-s − 41-s − 47-s − 15·48-s + 51-s + 57-s + 35·64-s − 67-s − 5·68-s + 69-s − 5·75-s − 5·76-s − 79-s − 83-s + 5·84-s − 5·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1307^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1307^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.680172055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.680172055\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 1307 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 3 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 7 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 17 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 19 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 23 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 37 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 41 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 47 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 67 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 79 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 83 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.13722118631882719409419950313, −5.86922743081577872904231196423, −5.74712404876748193306385024610, −5.73605491108148227668837320391, −5.62006404181632412294562886024, −5.20690834166375170573766711220, −4.99335970976818203948629035618, −4.86736008290012073606505039163, −4.76641050589906387140719162982, −4.30774589067694911108504902158, −4.20057657877478761353154296283, −3.73471482656569817659124516632, −3.48845620785917386631400538422, −3.47167376148172271993783476082, −3.24683583704902650781062792224, −2.92390178497797598580701023747, −2.89929838152777652369985804788, −2.64976638654147501120017849487, −2.45696394272249087291372283661, −2.35499636805085045409082402412, −1.96936033558421623595102299533, −1.67255858803533642187412778495, −1.56453416894178091232920346506, −1.17398477397839670073436279742, −1.02649999122370677179185768688,
1.02649999122370677179185768688, 1.17398477397839670073436279742, 1.56453416894178091232920346506, 1.67255858803533642187412778495, 1.96936033558421623595102299533, 2.35499636805085045409082402412, 2.45696394272249087291372283661, 2.64976638654147501120017849487, 2.89929838152777652369985804788, 2.92390178497797598580701023747, 3.24683583704902650781062792224, 3.47167376148172271993783476082, 3.48845620785917386631400538422, 3.73471482656569817659124516632, 4.20057657877478761353154296283, 4.30774589067694911108504902158, 4.76641050589906387140719162982, 4.86736008290012073606505039163, 4.99335970976818203948629035618, 5.20690834166375170573766711220, 5.62006404181632412294562886024, 5.73605491108148227668837320391, 5.74712404876748193306385024610, 5.86922743081577872904231196423, 6.13722118631882719409419950313
