| L(s) = 1 | + 3-s + 3.26·5-s − 4.21·7-s + 9-s + 1.64·11-s − 5.09·13-s + 3.26·15-s − 7.91·17-s − 2.28·19-s − 4.21·21-s + 2.37·23-s + 5.66·25-s + 27-s + 3.30·29-s + 6.02·31-s + 1.64·33-s − 13.7·35-s − 3.25·37-s − 5.09·39-s − 6.76·41-s + 0.995·43-s + 3.26·45-s − 7.03·47-s + 10.7·49-s − 7.91·51-s − 7.78·53-s + 5.38·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.46·5-s − 1.59·7-s + 0.333·9-s + 0.497·11-s − 1.41·13-s + 0.843·15-s − 1.91·17-s − 0.525·19-s − 0.918·21-s + 0.495·23-s + 1.13·25-s + 0.192·27-s + 0.613·29-s + 1.08·31-s + 0.287·33-s − 2.32·35-s − 0.535·37-s − 0.816·39-s − 1.05·41-s + 0.151·43-s + 0.486·45-s − 1.02·47-s + 1.53·49-s − 1.10·51-s − 1.06·53-s + 0.726·55-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 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 7.91T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 + 3.25T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 - 0.995T + 43T^{2} \) |
| 47 | \( 1 + 7.03T + 47T^{2} \) |
| 53 | \( 1 + 7.78T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 0.104T + 71T^{2} \) |
| 73 | \( 1 + 6.12T + 73T^{2} \) |
| 79 | \( 1 - 3.17T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 0.758T + 97T^{2} \) |
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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
−8.275707890780732237982267029327, −6.98961624480005355431184544252, −6.63406037146410006959657848205, −6.15211028265530472144269948501, −5.02043240214838772474515458337, −4.32100445522147252707512103048, −3.07390069788395908919158730894, −2.56232694150682437464519344108, −1.72149717070072404008678544923, 0,
1.72149717070072404008678544923, 2.56232694150682437464519344108, 3.07390069788395908919158730894, 4.32100445522147252707512103048, 5.02043240214838772474515458337, 6.15211028265530472144269948501, 6.63406037146410006959657848205, 6.98961624480005355431184544252, 8.275707890780732237982267029327
