| L(s) = 1 | + 2·5-s − 6·7-s − 5·13-s − 4·17-s − 8·19-s + 6·23-s + 5·25-s − 4·29-s − 14·31-s − 12·35-s − 2·37-s − 11·43-s + 6·47-s + 13·49-s − 2·53-s + 8·59-s − 7·61-s − 10·65-s − 3·67-s + 6·71-s − 9·73-s − 5·79-s − 32·83-s − 8·85-s + 12·89-s + 30·91-s − 16·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.26·7-s − 1.38·13-s − 0.970·17-s − 1.83·19-s + 1.25·23-s + 25-s − 0.742·29-s − 2.51·31-s − 2.02·35-s − 0.328·37-s − 1.67·43-s + 0.875·47-s + 13/7·49-s − 0.274·53-s + 1.04·59-s − 0.896·61-s − 1.24·65-s − 0.366·67-s + 0.712·71-s − 1.05·73-s − 0.562·79-s − 3.51·83-s − 0.867·85-s + 1.27·89-s + 3.14·91-s − 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.306269738621047060312420199241, −9.123510918964816904252867599255, −8.830959875282071263788795595127, −8.389507117795286092083415151965, −7.57127652442548273443255847358, −7.07357713572837024674482005907, −6.98157561107124714768255087562, −6.54569103784963785633081319823, −6.16052439956357683654385805772, −5.77744895464246993645296942924, −5.05026693910472787602972139594, −5.02030667800183179889151113607, −4.01025622547777093142113488831, −3.87207161599248744008655570664, −2.93071625228795596067591783021, −2.87165615338701617860984459457, −2.14386027160784954673923225528, −1.62436729480742711917682891759, 0, 0,
1.62436729480742711917682891759, 2.14386027160784954673923225528, 2.87165615338701617860984459457, 2.93071625228795596067591783021, 3.87207161599248744008655570664, 4.01025622547777093142113488831, 5.02030667800183179889151113607, 5.05026693910472787602972139594, 5.77744895464246993645296942924, 6.16052439956357683654385805772, 6.54569103784963785633081319823, 6.98157561107124714768255087562, 7.07357713572837024674482005907, 7.57127652442548273443255847358, 8.389507117795286092083415151965, 8.830959875282071263788795595127, 9.123510918964816904252867599255, 9.306269738621047060312420199241
