| L(s) = 1 | + 3-s + 2.53·5-s − 3.22·7-s + 9-s + 3.10·11-s − 6.06·13-s + 2.53·15-s − 5.94·17-s − 3.22·21-s − 6.71·23-s + 1.41·25-s + 27-s + 7.12·29-s + 7.63·31-s + 3.10·33-s − 8.17·35-s + 2.10·37-s − 6.06·39-s − 7.04·41-s − 9.36·43-s + 2.53·45-s − 4.65·47-s + 3.41·49-s − 5.94·51-s − 6.35·53-s + 7.86·55-s − 0.268·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·5-s − 1.21·7-s + 0.333·9-s + 0.936·11-s − 1.68·13-s + 0.653·15-s − 1.44·17-s − 0.704·21-s − 1.40·23-s + 0.282·25-s + 0.192·27-s + 1.32·29-s + 1.37·31-s + 0.540·33-s − 1.38·35-s + 0.346·37-s − 0.971·39-s − 1.09·41-s − 1.42·43-s + 0.377·45-s − 0.678·47-s + 0.487·49-s − 0.832·51-s − 0.872·53-s + 1.06·55-s − 0.0349·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 23 | \( 1 + 6.71T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 7.63T + 31T^{2} \) |
| 37 | \( 1 - 2.10T + 37T^{2} \) |
| 41 | \( 1 + 7.04T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 + 6.35T + 53T^{2} \) |
| 59 | \( 1 + 0.268T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 + 1.30T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 - 0.921T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 0.0864T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 3.32T + 97T^{2} \) |
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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
−8.167919246004900937174280761351, −7.04704844059734516184952481446, −6.52094437352877995551163102861, −6.11626791364696536639153149592, −4.90141789711959435266160939196, −4.27381946715023235967370966879, −3.14327371152579108624660100075, −2.46979851169930612728100428583, −1.69232083052172181852257790814, 0,
1.69232083052172181852257790814, 2.46979851169930612728100428583, 3.14327371152579108624660100075, 4.27381946715023235967370966879, 4.90141789711959435266160939196, 6.11626791364696536639153149592, 6.52094437352877995551163102861, 7.04704844059734516184952481446, 8.167919246004900937174280761351
