| L(s) = 1 | + 0.414·2-s − 1.82·4-s + 7-s − 1.58·8-s + 4.82·11-s − 0.828·13-s + 0.414·14-s + 3·16-s − 7.65·17-s − 2.82·19-s + 1.99·22-s + 3.65·23-s − 0.343·26-s − 1.82·28-s + 8·29-s + 8.48·31-s + 4.41·32-s − 3.17·34-s + 6·37-s − 1.17·38-s − 7.65·41-s − 1.65·43-s − 8.82·44-s + 1.51·46-s + 4·47-s + 49-s + 1.51·52-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s + 0.377·7-s − 0.560·8-s + 1.45·11-s − 0.229·13-s + 0.110·14-s + 0.750·16-s − 1.85·17-s − 0.648·19-s + 0.426·22-s + 0.762·23-s − 0.0672·26-s − 0.345·28-s + 1.48·29-s + 1.52·31-s + 0.780·32-s − 0.543·34-s + 0.986·37-s − 0.190·38-s − 1.19·41-s − 0.252·43-s − 1.33·44-s + 0.223·46-s + 0.583·47-s + 0.142·49-s + 0.210·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.626664267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626664267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−9.285349067717175880142989476131, −8.683523398064245931358573386170, −8.112904435866549713564142443748, −6.63093532373398252874187639222, −6.42487303242401377188897156611, −4.94815588590116147010231314792, −4.51655591641503877926536922441, −3.67706109614843316059969298026, −2.38002570622359334650757213469, −0.891019597558121091142297421618,
0.891019597558121091142297421618, 2.38002570622359334650757213469, 3.67706109614843316059969298026, 4.51655591641503877926536922441, 4.94815588590116147010231314792, 6.42487303242401377188897156611, 6.63093532373398252874187639222, 8.112904435866549713564142443748, 8.683523398064245931358573386170, 9.285349067717175880142989476131
