| L(s) = 1 | + 0.154·3-s + 0.882·5-s − 1.05·7-s − 2.97·9-s − 4.14·11-s − 3.65·13-s + 0.136·15-s + 4.60·17-s − 19-s − 0.163·21-s + 5.28·23-s − 4.22·25-s − 0.923·27-s + 4.39·29-s − 3.48·31-s − 0.641·33-s − 0.933·35-s − 3.30·37-s − 0.564·39-s + 7.84·41-s + 7.44·43-s − 2.62·45-s + 9.33·47-s − 5.88·49-s + 0.711·51-s − 53-s − 3.66·55-s + ⋯ |
| L(s) = 1 | + 0.0892·3-s + 0.394·5-s − 0.399·7-s − 0.992·9-s − 1.25·11-s − 1.01·13-s + 0.0352·15-s + 1.11·17-s − 0.229·19-s − 0.0356·21-s + 1.10·23-s − 0.844·25-s − 0.177·27-s + 0.816·29-s − 0.625·31-s − 0.111·33-s − 0.157·35-s − 0.542·37-s − 0.0904·39-s + 1.22·41-s + 1.13·43-s − 0.391·45-s + 1.36·47-s − 0.840·49-s + 0.0996·51-s − 0.137·53-s − 0.493·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.383964550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383964550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 0.154T + 3T^{2} \) |
| 5 | \( 1 - 0.882T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 + 3.65T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 - 4.39T + 29T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 - 7.44T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 - 0.201T + 67T^{2} \) |
| 71 | \( 1 - 5.78T + 71T^{2} \) |
| 73 | \( 1 - 8.99T + 73T^{2} \) |
| 79 | \( 1 + 1.88T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−8.357907276666603874885439643023, −7.73886043185639098610329311211, −7.09762029222121654241260288697, −6.06943709582382307426135566368, −5.45775055713222809095900208662, −4.93499206353758069912818891998, −3.68317319814247626475601896908, −2.78055100805470845120730052171, −2.28390771948878985433988559853, −0.63691271948589422907918726976,
0.63691271948589422907918726976, 2.28390771948878985433988559853, 2.78055100805470845120730052171, 3.68317319814247626475601896908, 4.93499206353758069912818891998, 5.45775055713222809095900208662, 6.06943709582382307426135566368, 7.09762029222121654241260288697, 7.73886043185639098610329311211, 8.357907276666603874885439643023
