| L(s) = 1 | + 2·5-s − 6·13-s + 2·17-s + 16·19-s + 8·23-s − 25-s + 10·37-s + 4·41-s + 8·43-s − 10·53-s − 24·59-s − 8·61-s − 12·65-s + 8·67-s − 10·73-s + 8·79-s − 16·83-s + 4·85-s + 32·95-s + 6·97-s + 24·101-s + 24·103-s + 24·107-s + 6·113-s + 16·115-s + 6·121-s − 12·125-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.66·13-s + 0.485·17-s + 3.67·19-s + 1.66·23-s − 1/5·25-s + 1.64·37-s + 0.624·41-s + 1.21·43-s − 1.37·53-s − 3.12·59-s − 1.02·61-s − 1.48·65-s + 0.977·67-s − 1.17·73-s + 0.900·79-s − 1.75·83-s + 0.433·85-s + 3.28·95-s + 0.609·97-s + 2.38·101-s + 2.36·103-s + 2.32·107-s + 0.564·113-s + 1.49·115-s + 6/11·121-s − 1.07·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.155023501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.155023501\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.573211195717052863458376747286, −9.559403810887046596194068800798, −9.134372311528488592581654300729, −8.727180656836095530643626803090, −7.81384812909713934384222367709, −7.62085269758916726756659700106, −7.33650315342164783247631140875, −7.23521676642155235816457860518, −6.30193014319975760955648090213, −5.93013041003012270490908461492, −5.70215169946733070720351632535, −5.09127233893193200574186638952, −4.66592509915856159382551282760, −4.62985023509734280906969009451, −3.34363961524202667958715621251, −3.21067899791441540781861482949, −2.76783635593910343706208450472, −2.10644436985628702607433223715, −1.30325039760591321006426536350, −0.802418142142576344592808250229,
0.802418142142576344592808250229, 1.30325039760591321006426536350, 2.10644436985628702607433223715, 2.76783635593910343706208450472, 3.21067899791441540781861482949, 3.34363961524202667958715621251, 4.62985023509734280906969009451, 4.66592509915856159382551282760, 5.09127233893193200574186638952, 5.70215169946733070720351632535, 5.93013041003012270490908461492, 6.30193014319975760955648090213, 7.23521676642155235816457860518, 7.33650315342164783247631140875, 7.62085269758916726756659700106, 7.81384812909713934384222367709, 8.727180656836095530643626803090, 9.134372311528488592581654300729, 9.559403810887046596194068800798, 9.573211195717052863458376747286
