| L(s) = 1 | − 5.14·3-s − 17.2·5-s − 32.9·7-s − 0.489·9-s + 39.2·11-s − 52.1·13-s + 88.9·15-s − 78.3·17-s + 24.5·19-s + 169.·21-s − 56.5·23-s + 173.·25-s + 141.·27-s − 192.·29-s + 125.·31-s − 202.·33-s + 569.·35-s + 6.44·37-s + 268.·39-s + 211.·41-s + 110.·43-s + 8.46·45-s − 522.·47-s + 742.·49-s + 403.·51-s + 125.·53-s − 677.·55-s + ⋯ |
| L(s) = 1 | − 0.990·3-s − 1.54·5-s − 1.77·7-s − 0.0181·9-s + 1.07·11-s − 1.11·13-s + 1.53·15-s − 1.11·17-s + 0.296·19-s + 1.76·21-s − 0.513·23-s + 1.38·25-s + 1.00·27-s − 1.23·29-s + 0.726·31-s − 1.06·33-s + 2.74·35-s + 0.0286·37-s + 1.10·39-s + 0.804·41-s + 0.390·43-s + 0.0280·45-s − 1.62·47-s + 2.16·49-s + 1.10·51-s + 0.324·53-s − 1.66·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - 251T \) |
| good | 3 | \( 1 + 5.14T + 27T^{2} \) |
| 5 | \( 1 + 17.2T + 125T^{2} \) |
| 7 | \( 1 + 32.9T + 343T^{2} \) |
| 11 | \( 1 - 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 192.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 6.44T + 5.06e4T^{2} \) |
| 41 | \( 1 - 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 522.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 115.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 996.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 93.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 633.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 73.0T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 348.T + 9.12e5T^{2} \) |
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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
−8.411979683144660460030181045135, −7.37497291944515097701076694161, −6.74314749642715719272609848371, −6.25573711645599153318201523310, −5.18497323118482460905139937564, −4.17103321652967805650586694665, −3.61148774843851340631224586684, −2.56575751281721363743706480799, −0.62923533396947842698674867012, 0,
0.62923533396947842698674867012, 2.56575751281721363743706480799, 3.61148774843851340631224586684, 4.17103321652967805650586694665, 5.18497323118482460905139937564, 6.25573711645599153318201523310, 6.74314749642715719272609848371, 7.37497291944515097701076694161, 8.411979683144660460030181045135
