L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 0.732·7-s − 8-s + 9-s + 3.73·11-s − 12-s − 2.73·13-s − 0.732·14-s + 16-s − 6.19·17-s − 18-s − 19-s − 0.732·21-s − 3.73·22-s + 5.92·23-s + 24-s + 2.73·26-s − 27-s + 0.732·28-s − 1.73·29-s − 2.46·31-s − 32-s − 3.73·33-s + 6.19·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.276·7-s − 0.353·8-s + 0.333·9-s + 1.12·11-s − 0.288·12-s − 0.757·13-s − 0.195·14-s + 0.250·16-s − 1.50·17-s − 0.235·18-s − 0.229·19-s − 0.159·21-s − 0.795·22-s + 1.23·23-s + 0.204·24-s + 0.535·26-s − 0.192·27-s + 0.138·28-s − 0.321·29-s − 0.442·31-s − 0.176·32-s − 0.649·33-s + 1.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 23 | \( 1 - 5.92T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 1.73T + 53T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 0.267T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 - 3.73T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.589637006218528387421201326961, −7.59551879178427233119901295986, −6.76204246059534951640029166572, −6.50337156095344798948573195491, −5.28864338973732717151319425623, −4.60046546227079517422315404741, −3.58457526565963067757022467920, −2.31132162016088562257512473367, −1.36647173771941590272376626945, 0,
1.36647173771941590272376626945, 2.31132162016088562257512473367, 3.58457526565963067757022467920, 4.60046546227079517422315404741, 5.28864338973732717151319425623, 6.50337156095344798948573195491, 6.76204246059534951640029166572, 7.59551879178427233119901295986, 8.589637006218528387421201326961