| L(s) = 1 | − 729·3-s − 1.48e4·5-s − 6.28e4·7-s + 5.31e5·9-s + 5.10e6·11-s + 1.14e7·13-s + 1.08e7·15-s + 1.19e8·17-s + 3.32e8·19-s + 4.58e7·21-s + 3.50e8·23-s − 1.00e9·25-s − 3.87e8·27-s − 1.76e9·29-s − 3.93e9·31-s − 3.72e9·33-s + 9.34e8·35-s − 7.80e9·37-s − 8.37e9·39-s + 5.28e10·41-s + 2.60e10·43-s − 7.89e9·45-s + 1.42e11·47-s − 9.29e10·49-s − 8.74e10·51-s + 1.37e10·53-s − 7.58e10·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.425·5-s − 0.202·7-s + 1/3·9-s + 0.868·11-s + 0.659·13-s + 0.245·15-s + 1.20·17-s + 1.62·19-s + 0.116·21-s + 0.494·23-s − 0.819·25-s − 0.192·27-s − 0.549·29-s − 0.796·31-s − 0.501·33-s + 0.0858·35-s − 0.500·37-s − 0.380·39-s + 1.73·41-s + 0.627·43-s − 0.141·45-s + 1.92·47-s − 0.959·49-s − 0.695·51-s + 0.0853·53-s − 0.369·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.451063567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.451063567\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{6} T \) |
| good | 5 | \( 1 + 594 p^{2} T + p^{13} T^{2} \) |
| 7 | \( 1 + 62896 T + p^{13} T^{2} \) |
| 11 | \( 1 - 464076 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 11484110 T + p^{13} T^{2} \) |
| 17 | \( 1 - 119964834 T + p^{13} T^{2} \) |
| 19 | \( 1 - 332601020 T + p^{13} T^{2} \) |
| 23 | \( 1 - 350924184 T + p^{13} T^{2} \) |
| 29 | \( 1 + 1761101946 T + p^{13} T^{2} \) |
| 31 | \( 1 + 3934224616 T + p^{13} T^{2} \) |
| 37 | \( 1 + 210907234 p T + p^{13} T^{2} \) |
| 41 | \( 1 - 52882647930 T + p^{13} T^{2} \) |
| 43 | \( 1 - 26018412164 T + p^{13} T^{2} \) |
| 47 | \( 1 - 142370739936 T + p^{13} T^{2} \) |
| 53 | \( 1 - 13770034398 T + p^{13} T^{2} \) |
| 59 | \( 1 - 336464984484 T + p^{13} T^{2} \) |
| 61 | \( 1 + 677260793938 T + p^{13} T^{2} \) |
| 67 | \( 1 - 262301598236 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1594961300520 T + p^{13} T^{2} \) |
| 73 | \( 1 - 578812819562 T + p^{13} T^{2} \) |
| 79 | \( 1 - 2495818789448 T + p^{13} T^{2} \) |
| 83 | \( 1 + 2693235578436 T + p^{13} T^{2} \) |
| 89 | \( 1 + 7935538832550 T + p^{13} T^{2} \) |
| 97 | \( 1 + 7858601662 T + p^{13} T^{2} \) |
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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
−16.82309709133038723126895389874, −15.73362297986268964319975753355, −14.08386428120255709766124678256, −12.35282825886615595741343488061, −11.20225329065044467753317984305, −9.458778835075893219839691226765, −7.45266564384342895502204967838, −5.70204145325414941548067396899, −3.69032400569397002434049259442, −1.01238310386308744570324330008,
1.01238310386308744570324330008, 3.69032400569397002434049259442, 5.70204145325414941548067396899, 7.45266564384342895502204967838, 9.458778835075893219839691226765, 11.20225329065044467753317984305, 12.35282825886615595741343488061, 14.08386428120255709766124678256, 15.73362297986268964319975753355, 16.82309709133038723126895389874
