| L(s) = 1 | + 2.03·2-s + 2.12·4-s − 1.29·5-s + 0.246·8-s − 2.63·10-s − 0.899·11-s − 1.69·13-s − 3.74·16-s + 17-s + 7.59·19-s − 2.74·20-s − 1.82·22-s + 2.48·23-s − 3.32·25-s − 3.44·26-s + 0.813·29-s + 8.78·31-s − 8.09·32-s + 2.03·34-s − 8.15·37-s + 15.4·38-s − 0.319·40-s + 9.01·41-s − 3.64·43-s − 1.90·44-s + 5.04·46-s + 4.49·47-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.06·4-s − 0.579·5-s + 0.0872·8-s − 0.832·10-s − 0.271·11-s − 0.470·13-s − 0.935·16-s + 0.242·17-s + 1.74·19-s − 0.614·20-s − 0.389·22-s + 0.518·23-s − 0.664·25-s − 0.675·26-s + 0.151·29-s + 1.57·31-s − 1.43·32-s + 0.348·34-s − 1.34·37-s + 2.49·38-s − 0.0505·40-s + 1.40·41-s − 0.555·43-s − 0.287·44-s + 0.743·46-s + 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.674879105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.674879105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 11 | \( 1 + 0.899T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 19 | \( 1 - 7.59T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 - 0.813T + 29T^{2} \) |
| 31 | \( 1 - 8.78T + 31T^{2} \) |
| 37 | \( 1 + 8.15T + 37T^{2} \) |
| 41 | \( 1 - 9.01T + 41T^{2} \) |
| 43 | \( 1 + 3.64T + 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 - 7.53T + 53T^{2} \) |
| 59 | \( 1 + 9.75T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 6.41T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 - 2.62T + 79T^{2} \) |
| 83 | \( 1 - 4.84T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−7.55769600142152781745840440270, −7.20578971473389420148115653590, −6.27073649842367739854829319355, −5.62109143679652927161594633196, −4.95187412803171524633244911328, −4.44966393969575495934619090683, −3.51237089818631883372296515283, −3.08713925267045490472661093364, −2.18092374777064041622970521152, −0.76102289098379256386397501392,
0.76102289098379256386397501392, 2.18092374777064041622970521152, 3.08713925267045490472661093364, 3.51237089818631883372296515283, 4.44966393969575495934619090683, 4.95187412803171524633244911328, 5.62109143679652927161594633196, 6.27073649842367739854829319355, 7.20578971473389420148115653590, 7.55769600142152781745840440270
