| L(s) = 1 | − 3-s + 3·5-s − 2·9-s − 3·15-s + 9·23-s + 4·25-s + 5·27-s + 5·31-s − 7·37-s − 6·45-s + 12·47-s − 7·49-s − 6·53-s − 15·59-s + 13·67-s − 9·69-s + 3·71-s − 4·75-s + 81-s − 9·89-s − 5·93-s + 17·97-s + 4·103-s + 7·111-s + 21·113-s + 27·115-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s − 0.774·15-s + 1.87·23-s + 4/5·25-s + 0.962·27-s + 0.898·31-s − 1.15·37-s − 0.894·45-s + 1.75·47-s − 49-s − 0.824·53-s − 1.95·59-s + 1.58·67-s − 1.08·69-s + 0.356·71-s − 0.461·75-s + 1/9·81-s − 0.953·89-s − 0.518·93-s + 1.72·97-s + 0.394·103-s + 0.664·111-s + 1.97·113-s + 2.51·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.047759332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047759332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.84163815829498118357633164477, −6.91210599271081945946951938107, −6.39367920775186201644515680583, −5.77148554369186015275320500685, −5.17363767447385357023112269188, −4.63920483570251043026902709830, −3.31610057639567574802422883219, −2.67536889552735053635478793091, −1.73800637486625496287263543940, −0.74716301316782723439191398619,
0.74716301316782723439191398619, 1.73800637486625496287263543940, 2.67536889552735053635478793091, 3.31610057639567574802422883219, 4.63920483570251043026902709830, 5.17363767447385357023112269188, 5.77148554369186015275320500685, 6.39367920775186201644515680583, 6.91210599271081945946951938107, 7.84163815829498118357633164477
