L(s) = 1 | − 2-s + 4-s + 3.12·7-s − 8-s + 2·11-s − 4·13-s − 3.12·14-s + 16-s − 3.12·17-s − 19-s − 2·22-s + 4·26-s + 3.12·28-s − 2·29-s + 9.12·31-s − 32-s + 3.12·34-s + 38-s − 5.12·41-s − 10.2·43-s + 2·44-s − 10.2·47-s + 2.75·49-s − 4·52-s + 4.24·53-s − 3.12·56-s + 2·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.18·7-s − 0.353·8-s + 0.603·11-s − 1.10·13-s − 0.834·14-s + 0.250·16-s − 0.757·17-s − 0.229·19-s − 0.426·22-s + 0.784·26-s + 0.590·28-s − 0.371·29-s + 1.63·31-s − 0.176·32-s + 0.535·34-s + 0.162·38-s − 0.800·41-s − 1.56·43-s + 0.301·44-s − 1.49·47-s + 0.393·49-s − 0.554·52-s + 0.583·53-s − 0.417·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 + 6.87T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.53065022155542769193869380124, −6.81855703903069503138138871376, −6.34078914137492780766939385891, −5.19136465650262397296598270495, −4.77562456258633642038820441409, −3.92329947105747106468742342945, −2.82231134717789269539524552329, −2.02033494488443966428932173692, −1.30240109854654402059735310272, 0,
1.30240109854654402059735310272, 2.02033494488443966428932173692, 2.82231134717789269539524552329, 3.92329947105747106468742342945, 4.77562456258633642038820441409, 5.19136465650262397296598270495, 6.34078914137492780766939385891, 6.81855703903069503138138871376, 7.53065022155542769193869380124