| L(s) = 1 | − 2-s − 3-s + 4-s − 0.414·5-s + 6-s − 8-s + 9-s + 0.414·10-s − 0.585·11-s − 12-s + 4.41·13-s + 0.414·15-s + 16-s − 1.17·17-s − 18-s − 6.24·19-s − 0.414·20-s + 0.585·22-s + 23-s + 24-s − 4.82·25-s − 4.41·26-s − 27-s − 4.07·29-s − 0.414·30-s − 0.585·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.185·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.130·10-s − 0.176·11-s − 0.288·12-s + 1.22·13-s + 0.106·15-s + 0.250·16-s − 0.284·17-s − 0.235·18-s − 1.43·19-s − 0.0926·20-s + 0.124·22-s + 0.208·23-s + 0.204·24-s − 0.965·25-s − 0.865·26-s − 0.192·27-s − 0.755·29-s − 0.0756·30-s − 0.105·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8307786674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8307786674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 0.414T + 5T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 0.585T + 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 43 | \( 1 - 2.65T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 9.31T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 6.24T + 73T^{2} \) |
| 79 | \( 1 + 5.89T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 + 3.75T + 89T^{2} \) |
| 97 | \( 1 + 4.65T + 97T^{2} \) |
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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
−8.128140606040497030816304062983, −7.19313455021916633954290814389, −6.71281019390530482365559993602, −5.87821888809636137722474008030, −5.44082560238324638830284831953, −4.17899827902401457297826891138, −3.77105958903934844202811457497, −2.48404101320642103369873874697, −1.65797372736151802930655624226, −0.53913811838849729580185174902,
0.53913811838849729580185174902, 1.65797372736151802930655624226, 2.48404101320642103369873874697, 3.77105958903934844202811457497, 4.17899827902401457297826891138, 5.44082560238324638830284831953, 5.87821888809636137722474008030, 6.71281019390530482365559993602, 7.19313455021916633954290814389, 8.128140606040497030816304062983
