| L(s) = 1 | − 0.442·2-s + 2.59·3-s − 1.80·4-s − 1.14·6-s − 1.77·7-s + 1.68·8-s + 3.75·9-s − 5.16·11-s − 4.68·12-s − 3.86·13-s + 0.784·14-s + 2.86·16-s − 5.69·17-s − 1.66·18-s − 1.54·19-s − 4.60·21-s + 2.28·22-s + 6.38·23-s + 4.37·24-s + 1.70·26-s + 1.96·27-s + 3.19·28-s + 3.55·29-s − 0.248·31-s − 4.63·32-s − 13.4·33-s + 2.52·34-s + ⋯ |
| L(s) = 1 | − 0.312·2-s + 1.50·3-s − 0.902·4-s − 0.469·6-s − 0.669·7-s + 0.594·8-s + 1.25·9-s − 1.55·11-s − 1.35·12-s − 1.07·13-s + 0.209·14-s + 0.716·16-s − 1.38·17-s − 0.391·18-s − 0.354·19-s − 1.00·21-s + 0.487·22-s + 1.33·23-s + 0.892·24-s + 0.334·26-s + 0.378·27-s + 0.604·28-s + 0.659·29-s − 0.0446·31-s − 0.818·32-s − 2.33·33-s + 0.432·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.381595166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381595166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.442T + 2T^{2} \) |
| 3 | \( 1 - 2.59T + 3T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 - 6.38T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 0.248T + 31T^{2} \) |
| 37 | \( 1 - 5.20T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 - 8.80T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.194913818572515288629285546449, −7.57065133944644414194782699926, −7.06347558017785586588289562521, −5.94106587330772535750398509469, −4.86168473801556559652954125060, −4.49656824958577474087989028741, −3.46238118379919470880099029264, −2.68958200814234590084267798866, −2.21827260164598141191331094642, −0.57261960915105736812628925995,
0.57261960915105736812628925995, 2.21827260164598141191331094642, 2.68958200814234590084267798866, 3.46238118379919470880099029264, 4.49656824958577474087989028741, 4.86168473801556559652954125060, 5.94106587330772535750398509469, 7.06347558017785586588289562521, 7.57065133944644414194782699926, 8.194913818572515288629285546449
