L(s) = 1 | + 2-s + 0.386·3-s + 4-s + 4.13·5-s + 0.386·6-s + 8-s − 2.85·9-s + 4.13·10-s − 2.54·11-s + 0.386·12-s + 3.74·13-s + 1.59·15-s + 16-s + 0.288·17-s − 2.85·18-s + 5.58·19-s + 4.13·20-s − 2.54·22-s − 7.65·23-s + 0.386·24-s + 12.1·25-s + 3.74·26-s − 2.25·27-s + 8.01·29-s + 1.59·30-s − 0.278·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.222·3-s + 0.5·4-s + 1.85·5-s + 0.157·6-s + 0.353·8-s − 0.950·9-s + 1.30·10-s − 0.768·11-s + 0.111·12-s + 1.03·13-s + 0.412·15-s + 0.250·16-s + 0.0699·17-s − 0.671·18-s + 1.28·19-s + 0.925·20-s − 0.543·22-s − 1.59·23-s + 0.0788·24-s + 2.42·25-s + 0.734·26-s − 0.434·27-s + 1.48·29-s + 0.291·30-s − 0.0500·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.638632300\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.638632300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 0.386T + 3T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 - 0.288T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 + 0.278T + 31T^{2} \) |
| 37 | \( 1 - 2.10T + 37T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 - 7.00T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 2.45T + 67T^{2} \) |
| 71 | \( 1 - 5.47T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 0.548T + 97T^{2} \) |
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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
−8.420609533151485750461388538395, −7.78036131212581019985506822065, −6.56667349798497049833928605588, −6.10933913667620273408661639987, −5.49508408722630753113838338078, −4.99765824948934980582410754556, −3.71770641003196476095412573697, −2.80312664376768221032454885743, −2.26696142706905656905519661075, −1.18387474310043389555355678874,
1.18387474310043389555355678874, 2.26696142706905656905519661075, 2.80312664376768221032454885743, 3.71770641003196476095412573697, 4.99765824948934980582410754556, 5.49508408722630753113838338078, 6.10933913667620273408661639987, 6.56667349798497049833928605588, 7.78036131212581019985506822065, 8.420609533151485750461388538395