| L(s) = 1 | − 0.414·2-s − 1.82·4-s − 1.41·7-s + 1.58·8-s − 6.24·11-s + 0.585·13-s + 0.585·14-s + 3·16-s + 6.82·17-s − 19-s + 2.58·22-s − 3.65·23-s − 0.242·26-s + 2.58·28-s + 1.41·29-s − 8.82·31-s − 4.41·32-s − 2.82·34-s + 0.585·37-s + 0.414·38-s − 8.24·41-s − 3.75·43-s + 11.4·44-s + 1.51·46-s + 3.65·47-s − 5·49-s − 1.07·52-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s − 0.534·7-s + 0.560·8-s − 1.88·11-s + 0.162·13-s + 0.156·14-s + 0.750·16-s + 1.65·17-s − 0.229·19-s + 0.551·22-s − 0.762·23-s − 0.0475·26-s + 0.488·28-s + 0.262·29-s − 1.58·31-s − 0.780·32-s − 0.485·34-s + 0.0963·37-s + 0.0671·38-s − 1.28·41-s − 0.572·43-s + 1.72·44-s + 0.223·46-s + 0.533·47-s − 0.714·49-s − 0.148·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6653533104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6653533104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 - 0.585T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 7.17T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.264103139920201003462964828883, −7.84648688006282962233995783039, −7.19554373802638507657587415396, −5.96891106224992177388979528260, −5.40624452326261759498993887652, −4.79326861007076927693510536819, −3.67669524627480762254298259072, −3.09777096013286524038510214081, −1.85819906304243172351811330659, −0.47413381271937967808859363874,
0.47413381271937967808859363874, 1.85819906304243172351811330659, 3.09777096013286524038510214081, 3.67669524627480762254298259072, 4.79326861007076927693510536819, 5.40624452326261759498993887652, 5.96891106224992177388979528260, 7.19554373802638507657587415396, 7.84648688006282962233995783039, 8.264103139920201003462964828883
