| L(s) = 1 | + 2-s + 2·3-s + 2·6-s + 5·7-s − 8-s + 3·9-s + 4·13-s + 5·14-s − 16-s + 3·17-s + 3·18-s − 8·19-s + 10·21-s + 9·23-s − 2·24-s + 4·26-s + 10·27-s − 12·29-s − 5·31-s + 3·34-s − 8·37-s − 8·38-s + 8·39-s − 6·41-s + 10·42-s − 20·43-s + 9·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.816·6-s + 1.88·7-s − 0.353·8-s + 9-s + 1.10·13-s + 1.33·14-s − 1/4·16-s + 0.727·17-s + 0.707·18-s − 1.83·19-s + 2.18·21-s + 1.87·23-s − 0.408·24-s + 0.784·26-s + 1.92·27-s − 2.22·29-s − 0.898·31-s + 0.514·34-s − 1.31·37-s − 1.29·38-s + 1.28·39-s − 0.937·41-s + 1.54·42-s − 3.04·43-s + 1.32·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.043973733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.043973733\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−11.64144106565294631226270546173, −11.24687569770698452644439309712, −10.85490003450880797361211485432, −10.50786472149861832151907270950, −9.922335082150076051493525082542, −9.115901610106276632386415197831, −8.691442805745142106916194205125, −8.593873130595189722389972798567, −8.152060182224736353905691858016, −7.46244244242676405754593605610, −7.07265906853716985985947101983, −6.48628705948829659225934705940, −5.66777547172578367733454129336, −5.15427063611177268782455829864, −4.70392693746425574624794487791, −4.17571636462357330071331152354, −3.44214755510998225275310245632, −3.10207045712088492456262227600, −1.76564402247031024152574226589, −1.68486586956382681209348921118,
1.68486586956382681209348921118, 1.76564402247031024152574226589, 3.10207045712088492456262227600, 3.44214755510998225275310245632, 4.17571636462357330071331152354, 4.70392693746425574624794487791, 5.15427063611177268782455829864, 5.66777547172578367733454129336, 6.48628705948829659225934705940, 7.07265906853716985985947101983, 7.46244244242676405754593605610, 8.152060182224736353905691858016, 8.593873130595189722389972798567, 8.691442805745142106916194205125, 9.115901610106276632386415197831, 9.922335082150076051493525082542, 10.50786472149861832151907270950, 10.85490003450880797361211485432, 11.24687569770698452644439309712, 11.64144106565294631226270546173
