| L(s) = 1 | − 4·4-s + 50·9-s + 114·11-s + 16·16-s − 38·19-s + 300·29-s + 64·31-s − 200·36-s − 516·41-s − 456·44-s − 275·49-s + 660·59-s − 26·61-s − 64·64-s + 1.28e3·71-s + 152·76-s + 1.40e3·79-s + 1.77e3·81-s + 1.20e3·89-s + 5.70e3·99-s + 2.12e3·101-s − 2.92e3·109-s − 1.20e3·116-s + 7.08e3·121-s − 256·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.85·9-s + 3.12·11-s + 1/4·16-s − 0.458·19-s + 1.92·29-s + 0.370·31-s − 0.925·36-s − 1.96·41-s − 1.56·44-s − 0.801·49-s + 1.45·59-s − 0.0545·61-s − 1/8·64-s + 2.14·71-s + 0.229·76-s + 1.99·79-s + 2.42·81-s + 1.42·89-s + 5.78·99-s + 2.09·101-s − 2.56·109-s − 0.960·116-s + 5.32·121-s − 0.185·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.587255328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.587255328\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 275 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 57 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )( 1 + 6 p T + p^{3} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5065 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 258 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 154525 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 127595 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 111130 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 330 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 131210 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 642 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 540865 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 700 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1143430 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 600 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 202430 T^{2} + p^{6} T^{4} \) |
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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
−9.644064037140311240561006068300, −9.546128984383800734137750149014, −9.161236716932795085658509517337, −8.673752622207650673948618893327, −8.194347193438798745591987146293, −7.957922448707580277618440977216, −6.97215459298714555403848428555, −6.91113085207857680330353429048, −6.48890752218509892622696000315, −6.36735309107665145319883590926, −5.48114803641410353242478146921, −4.67786630321155019324172541633, −4.65555582243016300614076263349, −4.07776084083849348591990500347, −3.60414352446249723225262212765, −3.37379483734888320502834456982, −2.13248276283789345642119631744, −1.70041086502442814865661357666, −1.02200080965530189226789347875, −0.78135078342163339690231969317,
0.78135078342163339690231969317, 1.02200080965530189226789347875, 1.70041086502442814865661357666, 2.13248276283789345642119631744, 3.37379483734888320502834456982, 3.60414352446249723225262212765, 4.07776084083849348591990500347, 4.65555582243016300614076263349, 4.67786630321155019324172541633, 5.48114803641410353242478146921, 6.36735309107665145319883590926, 6.48890752218509892622696000315, 6.91113085207857680330353429048, 6.97215459298714555403848428555, 7.957922448707580277618440977216, 8.194347193438798745591987146293, 8.673752622207650673948618893327, 9.161236716932795085658509517337, 9.546128984383800734137750149014, 9.644064037140311240561006068300
