| L(s) = 1 | − 0.185·3-s − 4.45·7-s − 2.96·9-s + 2.64·11-s + 1.30·13-s − 3.51·17-s − 19-s + 0.826·21-s − 6.52·23-s + 1.10·27-s − 5.20·29-s + 10.8·31-s − 0.490·33-s + 2.04·37-s − 0.241·39-s − 3.80·41-s + 4.77·43-s + 1.49·47-s + 12.8·49-s + 0.652·51-s + 0.225·53-s + 0.185·57-s − 2.86·59-s − 6.31·61-s + 13.2·63-s − 13.1·67-s + 1.21·69-s + ⋯ |
| L(s) = 1 | − 0.107·3-s − 1.68·7-s − 0.988·9-s + 0.797·11-s + 0.360·13-s − 0.853·17-s − 0.229·19-s + 0.180·21-s − 1.36·23-s + 0.212·27-s − 0.967·29-s + 1.94·31-s − 0.0854·33-s + 0.336·37-s − 0.0386·39-s − 0.593·41-s + 0.727·43-s + 0.217·47-s + 1.83·49-s + 0.0913·51-s + 0.0309·53-s + 0.0245·57-s − 0.372·59-s − 0.808·61-s + 1.66·63-s − 1.61·67-s + 0.145·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9259705789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9259705789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.185T + 3T^{2} \) |
| 7 | \( 1 + 4.45T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 - 0.225T + 53T^{2} \) |
| 59 | \( 1 + 2.86T + 59T^{2} \) |
| 61 | \( 1 + 6.31T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 3.84T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−8.659175646472029505471538419615, −7.79735849558926592157435627186, −6.77826896861966964435603104157, −6.16989474234417847559354766664, −5.93189348810692270618568733100, −4.59347935005849135591508890250, −3.76237981341429066971916659832, −3.06211752652064591989601183392, −2.13585591453109941088560000511, −0.53123580245396223865020900895,
0.53123580245396223865020900895, 2.13585591453109941088560000511, 3.06211752652064591989601183392, 3.76237981341429066971916659832, 4.59347935005849135591508890250, 5.93189348810692270618568733100, 6.16989474234417847559354766664, 6.77826896861966964435603104157, 7.79735849558926592157435627186, 8.659175646472029505471538419615
