L(s) = 1 | − 2-s − 3-s + 4-s − 2.13·5-s + 6-s + 0.0489·7-s − 8-s + 9-s + 2.13·10-s − 6.29·11-s − 12-s − 0.0489·14-s + 2.13·15-s + 16-s − 2.89·17-s − 18-s + 7.20·19-s − 2.13·20-s − 0.0489·21-s + 6.29·22-s + 2.71·23-s + 24-s − 0.432·25-s − 27-s + 0.0489·28-s + 4.91·29-s − 2.13·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.955·5-s + 0.408·6-s + 0.0184·7-s − 0.353·8-s + 0.333·9-s + 0.675·10-s − 1.89·11-s − 0.288·12-s − 0.0130·14-s + 0.551·15-s + 0.250·16-s − 0.700·17-s − 0.235·18-s + 1.65·19-s − 0.477·20-s − 0.0106·21-s + 1.34·22-s + 0.565·23-s + 0.204·24-s − 0.0865·25-s − 0.192·27-s + 0.00924·28-s + 0.912·29-s − 0.390·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5406647362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5406647362\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2.13T + 5T^{2} \) |
| 7 | \( 1 - 0.0489T + 7T^{2} \) |
| 11 | \( 1 + 6.29T + 11T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 - 4.91T + 29T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 - 0.176T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 - 6.71T + 43T^{2} \) |
| 47 | \( 1 + 7.20T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 - 7.10T + 61T^{2} \) |
| 67 | \( 1 + 5.38T + 67T^{2} \) |
| 71 | \( 1 - 8.71T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 - 2.47T + 97T^{2} \) |
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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
−10.00390947512864065102616632136, −9.186457851045932466113364521095, −8.046935348119268823976898860064, −7.63854300222723316824918697266, −6.85551515232118287896093746738, −5.58602573002298444695662775525, −4.90864646768007056794627738194, −3.57421431651013963229120400384, −2.42675244976133488697692612131, −0.62725804158590284016844894758,
0.62725804158590284016844894758, 2.42675244976133488697692612131, 3.57421431651013963229120400384, 4.90864646768007056794627738194, 5.58602573002298444695662775525, 6.85551515232118287896093746738, 7.63854300222723316824918697266, 8.046935348119268823976898860064, 9.186457851045932466113364521095, 10.00390947512864065102616632136