| L(s) = 1 | − 4.42e5·3-s + 2.22e7·5-s + 8.79e8·7-s + 1.01e11·9-s − 1.55e12·11-s − 1.97e12·13-s − 9.85e12·15-s − 2.09e14·17-s − 4.91e14·19-s − 3.88e14·21-s + 8.38e15·23-s − 1.14e16·25-s − 3.12e15·27-s − 9.20e16·29-s + 1.51e17·31-s + 6.86e17·33-s + 1.96e16·35-s − 1.71e17·37-s + 8.74e17·39-s − 4.58e17·41-s − 5.33e18·43-s + 2.25e18·45-s − 2.08e19·47-s − 2.65e19·49-s + 9.24e19·51-s − 5.59e19·53-s − 3.46e19·55-s + ⋯ |
| L(s) = 1 | − 1.44·3-s + 0.204·5-s + 0.168·7-s + 1.07·9-s − 1.64·11-s − 0.306·13-s − 0.294·15-s − 1.48·17-s − 0.968·19-s − 0.242·21-s + 1.83·23-s − 0.958·25-s − 0.108·27-s − 1.40·29-s + 1.07·31-s + 2.36·33-s + 0.0343·35-s − 0.158·37-s + 0.440·39-s − 0.129·41-s − 0.875·43-s + 0.219·45-s − 1.22·47-s − 0.971·49-s + 2.13·51-s − 0.828·53-s − 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.009989723632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.009989723632\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4.42e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 2.22e7T + 1.19e16T^{2} \) |
| 7 | \( 1 - 8.79e8T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.55e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.97e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.09e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 4.91e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 8.38e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 9.20e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.51e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.71e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 4.58e17T + 1.24e37T^{2} \) |
| 43 | \( 1 + 5.33e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.08e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 5.59e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 9.00e19T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.75e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 3.44e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 1.58e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 2.05e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + 1.20e22T + 4.42e43T^{2} \) |
| 83 | \( 1 + 2.39e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 4.69e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 6.79e22T + 4.96e45T^{2} \) |
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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
−10.90745149691471672054334730496, −9.921056665435802942556885494023, −8.469964998621577486518606399854, −7.16676351764302070216435139681, −6.19756539149563458517740011507, −5.18650844897429872276529125870, −4.56043402444213153077677689352, −2.77985238397191224564085117580, −1.62639369491093593778938463867, −0.04343594764978290103445316007,
0.04343594764978290103445316007, 1.62639369491093593778938463867, 2.77985238397191224564085117580, 4.56043402444213153077677689352, 5.18650844897429872276529125870, 6.19756539149563458517740011507, 7.16676351764302070216435139681, 8.469964998621577486518606399854, 9.921056665435802942556885494023, 10.90745149691471672054334730496
