| L(s) = 1 | + 9.26·3-s − 16.4·5-s + 34.5·7-s + 58.7·9-s + 7.42·11-s − 42.0·13-s − 151.·15-s − 59.9·19-s + 319.·21-s + 49.4·23-s + 144.·25-s + 294.·27-s + 259.·29-s + 92.2·31-s + 68.7·33-s − 566.·35-s − 207.·37-s − 389.·39-s + 176.·41-s + 19.0·43-s − 964.·45-s + 80.1·47-s + 847.·49-s − 319.·53-s − 121.·55-s − 554.·57-s − 11.0·59-s + ⋯ |
| L(s) = 1 | + 1.78·3-s − 1.46·5-s + 1.86·7-s + 2.17·9-s + 0.203·11-s − 0.896·13-s − 2.61·15-s − 0.723·19-s + 3.32·21-s + 0.448·23-s + 1.15·25-s + 2.09·27-s + 1.66·29-s + 0.534·31-s + 0.362·33-s − 2.73·35-s − 0.922·37-s − 1.59·39-s + 0.673·41-s + 0.0676·43-s − 3.19·45-s + 0.248·47-s + 2.47·49-s − 0.829·53-s − 0.298·55-s − 1.28·57-s − 0.0244·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.585284351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.585284351\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 9.26T + 27T^{2} \) |
| 5 | \( 1 + 16.4T + 125T^{2} \) |
| 7 | \( 1 - 34.5T + 343T^{2} \) |
| 11 | \( 1 - 7.42T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 59.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 92.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 19.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 80.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 319.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 11.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 712.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 484.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 443.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 337.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 840.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 456.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 638.T + 9.12e5T^{2} \) |
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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
−8.383673645374401439477524474209, −8.092538396032707970465601941824, −7.47607408072681807227666266274, −6.82675246407188353250830674047, −4.98083032279929021594218049224, −4.45755077488877367120737520326, −3.81956798037790329023128948733, −2.79531245915320433372789405926, −2.00738233534249110680674718523, −0.910659806303158143126390224310,
0.910659806303158143126390224310, 2.00738233534249110680674718523, 2.79531245915320433372789405926, 3.81956798037790329023128948733, 4.45755077488877367120737520326, 4.98083032279929021594218049224, 6.82675246407188353250830674047, 7.47607408072681807227666266274, 8.092538396032707970465601941824, 8.383673645374401439477524474209
