| L(s) = 1 | − 5·5-s + 4·7-s + 40·11-s − 90·13-s + 70·17-s − 40·19-s + 108·23-s + 25·25-s − 166·29-s + 40·31-s − 20·35-s − 130·37-s + 310·41-s + 268·43-s − 556·47-s − 327·49-s + 370·53-s − 200·55-s + 240·59-s − 130·61-s + 450·65-s − 876·67-s − 840·71-s + 250·73-s + 160·77-s + 880·79-s − 188·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.215·7-s + 1.09·11-s − 1.92·13-s + 0.998·17-s − 0.482·19-s + 0.979·23-s + 1/5·25-s − 1.06·29-s + 0.231·31-s − 0.0965·35-s − 0.577·37-s + 1.18·41-s + 0.950·43-s − 1.72·47-s − 0.953·49-s + 0.958·53-s − 0.490·55-s + 0.529·59-s − 0.272·61-s + 0.858·65-s − 1.59·67-s − 1.40·71-s + 0.400·73-s + 0.236·77-s + 1.25·79-s − 0.248·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 90 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 108 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 - 40 T + p^{3} T^{2} \) |
| 37 | \( 1 + 130 T + p^{3} T^{2} \) |
| 41 | \( 1 - 310 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 556 T + p^{3} T^{2} \) |
| 53 | \( 1 - 370 T + p^{3} T^{2} \) |
| 59 | \( 1 - 240 T + p^{3} T^{2} \) |
| 61 | \( 1 + 130 T + p^{3} T^{2} \) |
| 67 | \( 1 + 876 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 - 880 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1550 T + p^{3} T^{2} \) |
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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
−8.880435459045506637699282142544, −7.75149228741522212624482321301, −7.30976620124268879735740080411, −6.39726541743888768211038795511, −5.29154915522744304188234080363, −4.55236781774650986974686604953, −3.59587252137015897535299526699, −2.54113917767193055495123264266, −1.30669693221493768509300630899, 0,
1.30669693221493768509300630899, 2.54113917767193055495123264266, 3.59587252137015897535299526699, 4.55236781774650986974686604953, 5.29154915522744304188234080363, 6.39726541743888768211038795511, 7.30976620124268879735740080411, 7.75149228741522212624482321301, 8.880435459045506637699282142544
