| L(s) = 1 | − 2-s + 4-s + 1.09·5-s − 2.72·7-s − 8-s − 1.09·10-s − 4.35·11-s − 3.83·13-s + 2.72·14-s + 16-s − 4.52·17-s − 1.46·19-s + 1.09·20-s + 4.35·22-s − 2.47·23-s − 3.79·25-s + 3.83·26-s − 2.72·28-s + 9.77·29-s − 0.816·31-s − 32-s + 4.52·34-s − 2.99·35-s + 3.29·37-s + 1.46·38-s − 1.09·40-s − 6.03·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.490·5-s − 1.03·7-s − 0.353·8-s − 0.346·10-s − 1.31·11-s − 1.06·13-s + 0.729·14-s + 0.250·16-s − 1.09·17-s − 0.335·19-s + 0.245·20-s + 0.929·22-s − 0.517·23-s − 0.759·25-s + 0.752·26-s − 0.515·28-s + 1.81·29-s − 0.146·31-s − 0.176·32-s + 0.775·34-s − 0.505·35-s + 0.542·37-s + 0.237·38-s − 0.173·40-s − 0.942·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4396941619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4396941619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 + 3.83T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 9.77T + 29T^{2} \) |
| 31 | \( 1 + 0.816T + 31T^{2} \) |
| 37 | \( 1 - 3.29T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 - 2.88T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 9.19T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 6.62T + 97T^{2} \) |
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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
−7.71051129216511671164575362629, −7.33821378565279071781152827315, −6.27798405473050133635141232466, −6.16292383446923167743503995247, −5.04128388424012822620026697901, −4.40957219262180012473086552562, −3.13505107772201430650574322461, −2.58573154455468588403277514170, −1.87973679816920694651449836737, −0.33800035764872442640047898198,
0.33800035764872442640047898198, 1.87973679816920694651449836737, 2.58573154455468588403277514170, 3.13505107772201430650574322461, 4.40957219262180012473086552562, 5.04128388424012822620026697901, 6.16292383446923167743503995247, 6.27798405473050133635141232466, 7.33821378565279071781152827315, 7.71051129216511671164575362629
