| L(s) = 1 | − 1.57·2-s + 3.37·3-s + 0.466·4-s − 5.30·6-s + 2.36·7-s + 2.40·8-s + 8.42·9-s + 6.59·11-s + 1.57·12-s + 0.383·13-s − 3.71·14-s − 4.71·16-s − 3.37·17-s − 13.2·18-s − 6.43·19-s + 7.99·21-s − 10.3·22-s + 5.97·23-s + 8.14·24-s − 0.602·26-s + 18.3·27-s + 1.10·28-s − 4.55·29-s + 3.99·31-s + 2.58·32-s + 22.2·33-s + 5.30·34-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 1.95·3-s + 0.233·4-s − 2.16·6-s + 0.893·7-s + 0.851·8-s + 2.80·9-s + 1.98·11-s + 0.454·12-s + 0.106·13-s − 0.992·14-s − 1.17·16-s − 0.818·17-s − 3.11·18-s − 1.47·19-s + 1.74·21-s − 2.20·22-s + 1.24·23-s + 1.66·24-s − 0.118·26-s + 3.52·27-s + 0.208·28-s − 0.846·29-s + 0.716·31-s + 0.457·32-s + 3.88·33-s + 0.909·34-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
| \(L(1)\) |
\(\approx\) |
\(3.066038954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.066038954\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.57T + 2T^{2} \) |
| 3 | \( 1 - 3.37T + 3T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 - 6.59T + 11T^{2} \) |
| 13 | \( 1 - 0.383T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 + 6.43T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 + 4.55T + 29T^{2} \) |
| 31 | \( 1 - 3.99T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.40T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 + 4.77T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 - 6.30T + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 6.04T + 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 0.433T + 89T^{2} \) |
| 97 | \( 1 + 6.27T + 97T^{2} \) |
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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
−8.340951568395708213807494086053, −7.71553409853454332332681539680, −6.96051987661927403835022617384, −6.50545320651595901379872560832, −4.75331607827227555572603452163, −4.24677482926046401666348366271, −3.67542184959477523833416544886, −2.45059241257496034737692182004, −1.73258210071733254534733563847, −1.13587521934836189871290245792,
1.13587521934836189871290245792, 1.73258210071733254534733563847, 2.45059241257496034737692182004, 3.67542184959477523833416544886, 4.24677482926046401666348366271, 4.75331607827227555572603452163, 6.50545320651595901379872560832, 6.96051987661927403835022617384, 7.71553409853454332332681539680, 8.340951568395708213807494086053
