| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 5·7-s + 4·8-s − 2·9-s + 5·11-s + 3·12-s + 10·14-s + 5·16-s + 5·17-s − 4·18-s + 8·19-s + 5·21-s + 10·22-s + 4·23-s + 4·24-s − 2·27-s + 15·28-s + 3·29-s + 10·31-s + 6·32-s + 5·33-s + 10·34-s − 6·36-s − 11·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s + 1.88·7-s + 1.41·8-s − 2/3·9-s + 1.50·11-s + 0.866·12-s + 2.67·14-s + 5/4·16-s + 1.21·17-s − 0.942·18-s + 1.83·19-s + 1.09·21-s + 2.13·22-s + 0.834·23-s + 0.816·24-s − 0.384·27-s + 2.83·28-s + 0.557·29-s + 1.79·31-s + 1.06·32-s + 0.870·33-s + 1.71·34-s − 36-s − 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.75485825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.75485825\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 25 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 5 T + 11 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 57 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 101 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 17 T + 163 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 79 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 125 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 178 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 211 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 179 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 27 T + 373 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
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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
−7.954150554793730421546752127438, −7.61967275792665631910562103125, −7.13491348190458774624375751441, −6.85498103725003740869162639337, −6.75501272397092409204542361340, −6.10626727123943771171673673941, −5.56497929440396135110653108097, −5.54527113704774568523771377036, −5.17919711130896548246823913799, −4.88912277462681792457354010273, −4.40371117922923718394937413814, −4.11519216671259583496667582512, −3.61613676017403360747296827655, −3.47372446148312811818296624039, −2.81439664404922727967328455622, −2.69005565336692460095466983882, −2.16675829437366010754978288239, −1.49160499088345314488736973688, −1.07670236266960151342017963413, −1.05329000946807033464395479929,
1.05329000946807033464395479929, 1.07670236266960151342017963413, 1.49160499088345314488736973688, 2.16675829437366010754978288239, 2.69005565336692460095466983882, 2.81439664404922727967328455622, 3.47372446148312811818296624039, 3.61613676017403360747296827655, 4.11519216671259583496667582512, 4.40371117922923718394937413814, 4.88912277462681792457354010273, 5.17919711130896548246823913799, 5.54527113704774568523771377036, 5.56497929440396135110653108097, 6.10626727123943771171673673941, 6.75501272397092409204542361340, 6.85498103725003740869162639337, 7.13491348190458774624375751441, 7.61967275792665631910562103125, 7.954150554793730421546752127438
