| L(s) = 1 | + 4.41e3·2-s − 1.40e7·4-s − 2.10e11·8-s − 8.47e11·9-s − 9.50e12·11-s − 4.54e14·16-s − 3.73e15·18-s − 4.19e16·22-s − 9.56e16·23-s − 2.98e17·25-s − 3.80e18·29-s + 5.04e18·32-s + 1.19e19·36-s − 2.57e19·37-s + 3.01e20·43-s + 1.34e20·44-s − 4.21e20·46-s − 1.31e21·50-s − 5.53e20·53-s − 1.67e22·58-s + 3.75e22·64-s − 1.50e22·67-s − 3.60e22·71-s + 1.78e23·72-s − 1.13e23·74-s − 8.20e23·79-s + 7.17e23·81-s + ⋯ |
| L(s) = 1 | + 0.761·2-s − 0.420·4-s − 1.08·8-s − 9-s − 0.913·11-s − 0.403·16-s − 0.761·18-s − 0.695·22-s − 0.909·23-s − 25-s − 1.99·29-s + 0.774·32-s + 0.420·36-s − 0.643·37-s + 1.15·43-s + 0.383·44-s − 0.692·46-s − 0.761·50-s − 0.154·53-s − 1.52·58-s + 0.992·64-s − 0.224·67-s − 0.261·71-s + 1.08·72-s − 0.490·74-s − 1.56·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.6916787740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6916787740\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 4411 T + p^{25} T^{2} \) |
| 3 | \( 1 + p^{25} T^{2} \) |
| 5 | \( 1 + p^{25} T^{2} \) |
| 11 | \( 1 + 9505269254876 T + p^{25} T^{2} \) |
| 13 | \( 1 + p^{25} T^{2} \) |
| 17 | \( 1 + p^{25} T^{2} \) |
| 19 | \( 1 + p^{25} T^{2} \) |
| 23 | \( 1 + 95604060075370552 T + p^{25} T^{2} \) |
| 29 | \( 1 + 3807195347450164318 T + p^{25} T^{2} \) |
| 31 | \( 1 + p^{25} T^{2} \) |
| 37 | \( 1 + 25773142737840533286 T + p^{25} T^{2} \) |
| 41 | \( 1 + p^{25} T^{2} \) |
| 43 | \( 1 - \)\(30\!\cdots\!48\)\( T + p^{25} T^{2} \) |
| 47 | \( 1 + p^{25} T^{2} \) |
| 53 | \( 1 + \)\(55\!\cdots\!50\)\( T + p^{25} T^{2} \) |
| 59 | \( 1 + p^{25} T^{2} \) |
| 61 | \( 1 + p^{25} T^{2} \) |
| 67 | \( 1 + \)\(15\!\cdots\!36\)\( T + p^{25} T^{2} \) |
| 71 | \( 1 + \)\(36\!\cdots\!24\)\( T + p^{25} T^{2} \) |
| 73 | \( 1 + p^{25} T^{2} \) |
| 79 | \( 1 + \)\(82\!\cdots\!32\)\( T + p^{25} T^{2} \) |
| 83 | \( 1 + p^{25} T^{2} \) |
| 89 | \( 1 + p^{25} T^{2} \) |
| 97 | \( 1 + p^{25} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.11039969697398311648508766443, −9.755533693000087180886214285247, −8.687009297585923205442858813315, −7.63818198306635098999317389545, −5.94063844068593959646742885429, −5.40443014363834424447037537226, −4.15111986897859115444396668047, −3.17563953721982532172413688662, −2.09290403092636676961483912322, −0.29300586442806956321829653294,
0.29300586442806956321829653294, 2.09290403092636676961483912322, 3.17563953721982532172413688662, 4.15111986897859115444396668047, 5.40443014363834424447037537226, 5.94063844068593959646742885429, 7.63818198306635098999317389545, 8.687009297585923205442858813315, 9.755533693000087180886214285247, 11.11039969697398311648508766443
