| L(s) = 1 | − 3-s + 4·5-s − 2·11-s + 12·13-s − 4·15-s − 4·17-s − 4·19-s − 2·23-s + 5·25-s + 27-s − 4·29-s + 2·33-s − 2·37-s − 12·39-s − 8·43-s + 12·47-s + 4·51-s + 6·53-s − 8·55-s + 4·57-s − 8·59-s + 6·61-s + 48·65-s + 8·67-s + 2·69-s + 28·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.603·11-s + 3.32·13-s − 1.03·15-s − 0.970·17-s − 0.917·19-s − 0.417·23-s + 25-s + 0.192·27-s − 0.742·29-s + 0.348·33-s − 0.328·37-s − 1.92·39-s − 1.21·43-s + 1.75·47-s + 0.560·51-s + 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.04·59-s + 0.768·61-s + 5.95·65-s + 0.977·67-s + 0.240·69-s + 3.32·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208203587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208203587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.83553250140472189725151166527, −10.45008187374833463644346171798, −10.31919960158553618696624186020, −9.399908899577025617190880326426, −9.393406871908638830482608094938, −8.632588966772118602689620141827, −8.438036275958671216387069985601, −8.070712938542017047973833027263, −7.10718021291927976537269469928, −6.59240410576927878018585832543, −6.33009998602463648975077156395, −5.76722286330627082202851542354, −5.73306790284993074620557280429, −5.12712483830821929972538849272, −4.26749392543257546348208727425, −3.82794037451527102611794146884, −3.19062712664587901307123999110, −2.01887921111433387468975441929, −2.00116482784921729319586394075, −0.907627878263744397577862395914,
0.907627878263744397577862395914, 2.00116482784921729319586394075, 2.01887921111433387468975441929, 3.19062712664587901307123999110, 3.82794037451527102611794146884, 4.26749392543257546348208727425, 5.12712483830821929972538849272, 5.73306790284993074620557280429, 5.76722286330627082202851542354, 6.33009998602463648975077156395, 6.59240410576927878018585832543, 7.10718021291927976537269469928, 8.070712938542017047973833027263, 8.438036275958671216387069985601, 8.632588966772118602689620141827, 9.393406871908638830482608094938, 9.399908899577025617190880326426, 10.31919960158553618696624186020, 10.45008187374833463644346171798, 10.83553250140472189725151166527
