| L(s) = 1 | + 0.963·2-s + 3-s − 1.07·4-s + 5-s + 0.963·6-s + 2.92·7-s − 2.95·8-s + 9-s + 0.963·10-s + 0.188·11-s − 1.07·12-s + 0.702·13-s + 2.82·14-s + 15-s − 0.705·16-s − 6.37·17-s + 0.963·18-s − 0.665·19-s − 1.07·20-s + 2.92·21-s + 0.181·22-s − 2.95·24-s + 25-s + 0.676·26-s + 27-s − 3.14·28-s + 3.25·29-s + ⋯ |
| L(s) = 1 | + 0.681·2-s + 0.577·3-s − 0.536·4-s + 0.447·5-s + 0.393·6-s + 1.10·7-s − 1.04·8-s + 0.333·9-s + 0.304·10-s + 0.0568·11-s − 0.309·12-s + 0.194·13-s + 0.753·14-s + 0.258·15-s − 0.176·16-s − 1.54·17-s + 0.227·18-s − 0.152·19-s − 0.239·20-s + 0.639·21-s + 0.0387·22-s − 0.604·24-s + 0.200·25-s + 0.132·26-s + 0.192·27-s − 0.593·28-s + 0.603·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.644539600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.644539600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.963T + 2T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 - 0.188T + 11T^{2} \) |
| 13 | \( 1 - 0.702T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 + 0.665T + 19T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 3.44T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 + 2.27T + 53T^{2} \) |
| 59 | \( 1 + 0.830T + 59T^{2} \) |
| 61 | \( 1 - 9.81T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 - 0.749T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 8.28T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.105295405563843922948887905894, −7.07129233128746512263506449266, −6.36640752249592780049094833863, −5.59366061136235144812488120406, −4.88870568760349139569769165097, −4.32552995396003647274756023861, −3.74670254216660388344729908809, −2.62468187294715290977774098188, −2.05807388503308786495019363768, −0.844684785860145858880450154760,
0.844684785860145858880450154760, 2.05807388503308786495019363768, 2.62468187294715290977774098188, 3.74670254216660388344729908809, 4.32552995396003647274756023861, 4.88870568760349139569769165097, 5.59366061136235144812488120406, 6.36640752249592780049094833863, 7.07129233128746512263506449266, 8.105295405563843922948887905894
