| L(s) = 1 | − 3-s + 5-s − 3.82·7-s + 9-s + 1.41·11-s + 3.41·13-s − 15-s + 0.414·17-s − 4.58·19-s + 3.82·21-s − 23-s + 25-s − 27-s − 6.41·29-s − 31-s − 1.41·33-s − 3.82·35-s + 9.48·37-s − 3.41·39-s − 1.24·41-s + 3.65·43-s + 45-s + 2.24·47-s + 7.65·49-s − 0.414·51-s − 3.24·53-s + 1.41·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.44·7-s + 0.333·9-s + 0.426·11-s + 0.946·13-s − 0.258·15-s + 0.100·17-s − 1.05·19-s + 0.835·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.19·29-s − 0.179·31-s − 0.246·33-s − 0.647·35-s + 1.55·37-s − 0.546·39-s − 0.194·41-s + 0.557·43-s + 0.149·45-s + 0.327·47-s + 1.09·49-s − 0.0580·51-s − 0.445·53-s + 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 29 | \( 1 + 6.41T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 9.48T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 1.48T + 67T^{2} \) |
| 71 | \( 1 - 0.0710T + 71T^{2} \) |
| 73 | \( 1 + 6.24T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.595937467135021247914414959312, −7.53589422766108682550872043048, −6.64981110830050194207613275425, −6.13032738138183386173152007094, −5.67839630902303545634662909309, −4.37509914595426252811672555586, −3.68367672303531482370159330480, −2.66702214033169657853567259540, −1.40366706460576317196449326817, 0,
1.40366706460576317196449326817, 2.66702214033169657853567259540, 3.68367672303531482370159330480, 4.37509914595426252811672555586, 5.67839630902303545634662909309, 6.13032738138183386173152007094, 6.64981110830050194207613275425, 7.53589422766108682550872043048, 8.595937467135021247914414959312
