L(s) = 1 | − 3·3-s + 5.67·5-s − 22.9·7-s + 9·9-s + 32.5·11-s − 13·13-s − 17.0·15-s + 107.·17-s − 116.·19-s + 68.9·21-s + 65.1·23-s − 92.7·25-s − 27·27-s + 92.5·29-s − 51.1·31-s − 97.7·33-s − 130.·35-s + 267.·37-s + 39·39-s − 392.·41-s − 317.·43-s + 51.0·45-s + 114.·47-s + 184.·49-s − 323.·51-s − 618·53-s + 184.·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.507·5-s − 1.24·7-s + 0.333·9-s + 0.893·11-s − 0.277·13-s − 0.293·15-s + 1.53·17-s − 1.41·19-s + 0.716·21-s + 0.591·23-s − 0.742·25-s − 0.192·27-s + 0.592·29-s − 0.296·31-s − 0.515·33-s − 0.629·35-s + 1.18·37-s + 0.160·39-s − 1.49·41-s − 1.12·43-s + 0.169·45-s + 0.356·47-s + 0.538·49-s − 0.888·51-s − 1.60·53-s + 0.453·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
good | 5 | \( 1 - 5.67T + 125T^{2} \) |
| 7 | \( 1 + 22.9T + 343T^{2} \) |
| 11 | \( 1 - 32.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 65.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 92.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 51.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 392.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 317.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 618T + 1.48e5T^{2} \) |
| 59 | \( 1 - 600.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 857.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 422.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 428.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 235.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 64.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 703.T + 9.12e5T^{2} \) |
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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
−9.841839921142832828255431859017, −9.179178020326735357021471422436, −7.973570615405222832461378867864, −6.71863398337458049635345465979, −6.29204001387263007178699800388, −5.33068767524713113108601103044, −4.06917848371804919548889850477, −2.99810341026911747684433261168, −1.46581745133449907581935018549, 0,
1.46581745133449907581935018549, 2.99810341026911747684433261168, 4.06917848371804919548889850477, 5.33068767524713113108601103044, 6.29204001387263007178699800388, 6.71863398337458049635345465979, 7.973570615405222832461378867864, 9.179178020326735357021471422436, 9.841839921142832828255431859017