| L(s) = 1 | + 2.11·2-s + 2.47·4-s + 1.31·7-s + 1.01·8-s + 5.39·11-s − 1.43·13-s + 2.77·14-s − 2.81·16-s + 6.75·17-s − 1.39·19-s + 11.4·22-s − 3.17·23-s − 3.03·26-s + 3.24·28-s + 5.09·29-s + 31-s − 7.98·32-s + 14.2·34-s + 10.2·37-s − 2.95·38-s − 5.05·41-s + 11.5·43-s + 13.3·44-s − 6.71·46-s − 10.6·47-s − 5.27·49-s − 3.55·52-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.23·4-s + 0.495·7-s + 0.357·8-s + 1.62·11-s − 0.398·13-s + 0.741·14-s − 0.703·16-s + 1.63·17-s − 0.320·19-s + 2.43·22-s − 0.661·23-s − 0.595·26-s + 0.614·28-s + 0.946·29-s + 0.179·31-s − 1.41·32-s + 2.45·34-s + 1.69·37-s − 0.480·38-s − 0.789·41-s + 1.75·43-s + 2.01·44-s − 0.990·46-s − 1.55·47-s − 0.754·49-s − 0.493·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.762238221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.762238221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 5.39T + 11T^{2} \) |
| 13 | \( 1 + 1.43T + 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 + 0.540T + 71T^{2} \) |
| 73 | \( 1 - 8.59T + 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 - 0.00535T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 - 1.80T + 97T^{2} \) |
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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
−7.85371817841238312208307033801, −6.95523535831555148361039475139, −6.30855380297190029065965854590, −5.82291783471477570876180558966, −4.95022732017751279127494765936, −4.39862572549706757765313275757, −3.71928438129244349027585566999, −3.04612353522956854906354674877, −2.04290527871098955110532496871, −1.03781942199209279739036929220,
1.03781942199209279739036929220, 2.04290527871098955110532496871, 3.04612353522956854906354674877, 3.71928438129244349027585566999, 4.39862572549706757765313275757, 4.95022732017751279127494765936, 5.82291783471477570876180558966, 6.30855380297190029065965854590, 6.95523535831555148361039475139, 7.85371817841238312208307033801
