| L(s) = 1 | + 2.80·3-s + 0.563·5-s − 3.54·7-s + 4.87·9-s + 0.185·11-s + 1.50·13-s + 1.58·15-s − 4.26·17-s − 5.87·19-s − 9.94·21-s − 4.13·23-s − 4.68·25-s + 5.25·27-s + 6.53·29-s − 9.18·31-s + 0.519·33-s − 1.99·35-s + 2.36·37-s + 4.23·39-s − 1.06·41-s + 9.99·43-s + 2.74·45-s + 6.05·47-s + 5.56·49-s − 11.9·51-s − 6.60·53-s + 0.104·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.251·5-s − 1.34·7-s + 1.62·9-s + 0.0558·11-s + 0.418·13-s + 0.408·15-s − 1.03·17-s − 1.34·19-s − 2.17·21-s − 0.862·23-s − 0.936·25-s + 1.01·27-s + 1.21·29-s − 1.65·31-s + 0.0904·33-s − 0.337·35-s + 0.389·37-s + 0.677·39-s − 0.166·41-s + 1.52·43-s + 0.409·45-s + 0.883·47-s + 0.795·49-s − 1.67·51-s − 0.907·53-s + 0.0140·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 - 0.563T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 - 0.185T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 - 6.53T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 - 6.05T + 47T^{2} \) |
| 53 | \( 1 + 6.60T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 6.98T + 61T^{2} \) |
| 67 | \( 1 + 8.95T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 5.71T + 97T^{2} \) |
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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
−7.79524960414527317252735186801, −7.14764148194007388423404939761, −6.31427336187892105191909462226, −5.89521832990292879651398838860, −4.35338730566431136892912739339, −3.98116728875870514251600090013, −3.11451778025107588054251625336, −2.45033744260081653039387558470, −1.73129865917974765233080451338, 0,
1.73129865917974765233080451338, 2.45033744260081653039387558470, 3.11451778025107588054251625336, 3.98116728875870514251600090013, 4.35338730566431136892912739339, 5.89521832990292879651398838860, 6.31427336187892105191909462226, 7.14764148194007388423404939761, 7.79524960414527317252735186801
