L(s) = 1 | − 0.895·2-s + 0.415·3-s − 1.19·4-s − 0.372·6-s − 2.78·7-s + 2.86·8-s − 2.82·9-s − 5.25·11-s − 0.498·12-s − 1.97·13-s + 2.49·14-s − 0.168·16-s − 6.98·17-s + 2.53·18-s + 8.09·19-s − 1.15·21-s + 4.70·22-s − 3.31·23-s + 1.19·24-s + 1.76·26-s − 2.42·27-s + 3.33·28-s + 1.04·29-s − 3.07·31-s − 5.57·32-s − 2.18·33-s + 6.25·34-s + ⋯ |
L(s) = 1 | − 0.633·2-s + 0.240·3-s − 0.599·4-s − 0.152·6-s − 1.05·7-s + 1.01·8-s − 0.942·9-s − 1.58·11-s − 0.143·12-s − 0.546·13-s + 0.666·14-s − 0.0420·16-s − 1.69·17-s + 0.596·18-s + 1.85·19-s − 0.252·21-s + 1.00·22-s − 0.690·23-s + 0.243·24-s + 0.346·26-s − 0.466·27-s + 0.630·28-s + 0.193·29-s − 0.553·31-s − 0.985·32-s − 0.380·33-s + 1.07·34-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
\(L(1)\) |
\(\approx\) |
\(0.01458051689\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01458051689\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.895T + 2T^{2} \) |
| 3 | \( 1 - 0.415T + 3T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 + 6.98T + 17T^{2} \) |
| 19 | \( 1 - 8.09T + 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 + 6.04T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 + 9.07T + 61T^{2} \) |
| 67 | \( 1 + 8.20T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 + 8.17T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.14T + 97T^{2} \) |
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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
−8.180102679030258740527083895571, −7.52589938143718325591038842957, −6.89657898404643544832586821926, −5.85924451323437723378101954282, −5.20815010384748925319216221993, −4.56936792584368426100990341658, −3.36300028567196030878590874969, −2.88465415216417112638280187378, −1.82838294505498075965953899879, −0.06509704169393321059836496555,
0.06509704169393321059836496555, 1.82838294505498075965953899879, 2.88465415216417112638280187378, 3.36300028567196030878590874969, 4.56936792584368426100990341658, 5.20815010384748925319216221993, 5.85924451323437723378101954282, 6.89657898404643544832586821926, 7.52589938143718325591038842957, 8.180102679030258740527083895571