L(s) = 1 | − 3-s − 4.20·5-s − 3.93·7-s + 9-s + 2.04·11-s + 2.48·13-s + 4.20·15-s − 7.89·17-s + 4.59·19-s + 3.93·21-s − 3.05·23-s + 12.6·25-s − 27-s + 10.2·29-s + 4.68·31-s − 2.04·33-s + 16.5·35-s + 9.76·37-s − 2.48·39-s − 7.20·41-s − 4.73·43-s − 4.20·45-s + 8.18·47-s + 8.47·49-s + 7.89·51-s − 6.41·53-s − 8.57·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.87·5-s − 1.48·7-s + 0.333·9-s + 0.615·11-s + 0.689·13-s + 1.08·15-s − 1.91·17-s + 1.05·19-s + 0.858·21-s − 0.636·23-s + 2.53·25-s − 0.192·27-s + 1.90·29-s + 0.842·31-s − 0.355·33-s + 2.79·35-s + 1.60·37-s − 0.398·39-s − 1.12·41-s − 0.721·43-s − 0.626·45-s + 1.19·47-s + 1.21·49-s + 1.10·51-s − 0.881·53-s − 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 19 | \( 1 - 4.59T + 19T^{2} \) |
| 23 | \( 1 + 3.05T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 9.76T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 - 8.18T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 + 0.838T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 8.17T + 83T^{2} \) |
| 89 | \( 1 - 9.03T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.077710805324539861121359519266, −7.23285937716270344966159037876, −6.40898736183490837901708359291, −6.36553288681513367658049621353, −4.78758040777313821496565882917, −4.25537957446811166711745220030, −3.52104785591883713288368326954, −2.80469873886323260491954814055, −0.945475395247190560206654767374, 0,
0.945475395247190560206654767374, 2.80469873886323260491954814055, 3.52104785591883713288368326954, 4.25537957446811166711745220030, 4.78758040777313821496565882917, 6.36553288681513367658049621353, 6.40898736183490837901708359291, 7.23285937716270344966159037876, 8.077710805324539861121359519266