| L(s) = 1 | + 1.83·3-s − 3.36·5-s − 0.752·7-s + 0.354·9-s + 6.48·11-s + 2.38·13-s − 6.15·15-s − 3.19·17-s + 3.80·19-s − 1.37·21-s + 2.77·23-s + 6.29·25-s − 4.84·27-s + 0.633·29-s − 3.86·31-s + 11.8·33-s + 2.52·35-s + 0.458·37-s + 4.36·39-s − 1.11·41-s + 4.41·43-s − 1.19·45-s − 11.5·47-s − 6.43·49-s − 5.85·51-s + 0.881·53-s − 21.7·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s − 1.50·5-s − 0.284·7-s + 0.118·9-s + 1.95·11-s + 0.660·13-s − 1.58·15-s − 0.775·17-s + 0.872·19-s − 0.300·21-s + 0.577·23-s + 1.25·25-s − 0.932·27-s + 0.117·29-s − 0.694·31-s + 2.06·33-s + 0.427·35-s + 0.0753·37-s + 0.698·39-s − 0.174·41-s + 0.674·43-s − 0.177·45-s − 1.67·47-s − 0.919·49-s − 0.820·51-s + 0.121·53-s − 2.93·55-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
| \(L(1)\) |
\(\approx\) |
\(2.254287238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.254287238\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.83T + 3T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 7 | \( 1 + 0.752T + 7T^{2} \) |
| 11 | \( 1 - 6.48T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 - 0.633T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 0.458T + 37T^{2} \) |
| 41 | \( 1 + 1.11T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 0.881T + 53T^{2} \) |
| 59 | \( 1 - 3.24T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 - 0.176T + 71T^{2} \) |
| 73 | \( 1 - 5.58T + 73T^{2} \) |
| 79 | \( 1 + 0.513T + 79T^{2} \) |
| 83 | \( 1 - 9.76T + 83T^{2} \) |
| 89 | \( 1 - 1.01T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.204712140342824786610242257355, −7.45237209165417855531588459662, −6.83276516745500641083535437324, −6.20019274304130098652930820630, −4.97625464094753828285988007631, −4.05360033875965487581510500111, −3.61051843700839269192202776066, −3.14930132136529828483837493599, −1.88849331705094467007573682138, −0.76269096330960179041951449243,
0.76269096330960179041951449243, 1.88849331705094467007573682138, 3.14930132136529828483837493599, 3.61051843700839269192202776066, 4.05360033875965487581510500111, 4.97625464094753828285988007631, 6.20019274304130098652930820630, 6.83276516745500641083535437324, 7.45237209165417855531588459662, 8.204712140342824786610242257355
