L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 4.47·13-s − 2·15-s + 4.47·17-s − 2.47·19-s + 21-s + 23-s − 25-s − 27-s − 2·29-s + 2.47·31-s − 2·35-s + 6.94·37-s + 4.47·39-s − 6·41-s + 4.94·43-s + 2·45-s + 2.47·47-s + 49-s − 4.47·51-s − 10.9·53-s + 2.47·57-s − 4·59-s − 6·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 0.333·9-s − 1.24·13-s − 0.516·15-s + 1.08·17-s − 0.567·19-s + 0.218·21-s + 0.208·23-s − 0.200·25-s − 0.192·27-s − 0.371·29-s + 0.444·31-s − 0.338·35-s + 1.14·37-s + 0.716·39-s − 0.937·41-s + 0.753·43-s + 0.298·45-s + 0.360·47-s + 0.142·49-s − 0.626·51-s − 1.50·53-s + 0.327·57-s − 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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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
−7.57217738192382147009034730617, −6.59887555103524585392383492632, −6.17820793698472085055739435832, −5.42865654179937754919858830837, −4.89248644221476904982666450124, −4.01664712670402337909545823196, −2.98715823273683233786617758464, −2.22893314956303996702896408026, −1.25723939815807264195696589803, 0,
1.25723939815807264195696589803, 2.22893314956303996702896408026, 2.98715823273683233786617758464, 4.01664712670402337909545823196, 4.89248644221476904982666450124, 5.42865654179937754919858830837, 6.17820793698472085055739435832, 6.59887555103524585392383492632, 7.57217738192382147009034730617