| L(s) = 1 | + 1.74·2-s − 0.766·3-s + 1.03·4-s − 1.33·6-s + 2.32·7-s − 1.68·8-s − 2.41·9-s − 4.04·11-s − 0.791·12-s − 3.88·13-s + 4.05·14-s − 4.99·16-s + 4.16·17-s − 4.20·18-s − 1.78·21-s − 7.03·22-s + 7.90·23-s + 1.29·24-s − 6.75·26-s + 4.14·27-s + 2.40·28-s − 7.12·29-s − 2.72·31-s − 5.33·32-s + 3.09·33-s + 7.25·34-s − 2.49·36-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 0.442·3-s + 0.516·4-s − 0.544·6-s + 0.879·7-s − 0.595·8-s − 0.804·9-s − 1.21·11-s − 0.228·12-s − 1.07·13-s + 1.08·14-s − 1.24·16-s + 1.01·17-s − 0.990·18-s − 0.388·21-s − 1.50·22-s + 1.64·23-s + 0.263·24-s − 1.32·26-s + 0.798·27-s + 0.453·28-s − 1.32·29-s − 0.488·31-s − 0.943·32-s + 0.539·33-s + 1.24·34-s − 0.415·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237960261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237960261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.74T + 2T^{2} \) |
| 3 | \( 1 + 0.766T + 3T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 23 | \( 1 - 7.90T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 + 2.72T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3.50T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 + 0.333T + 47T^{2} \) |
| 53 | \( 1 + 6.90T + 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 + 1.35T + 61T^{2} \) |
| 67 | \( 1 - 7.79T + 67T^{2} \) |
| 71 | \( 1 + 0.0586T + 71T^{2} \) |
| 73 | \( 1 - 9.15T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 3.84T + 97T^{2} \) |
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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
−7.59123761733401727333375368453, −7.00270559397515197426920825528, −5.94094495435168887474430200565, −5.44722087764392814401734091744, −5.05230414477415383945981497923, −4.55650226276854385696390888979, −3.43992235223035630666817469070, −2.84698544346114256596719804471, −2.08573337518142273594511285902, −0.58364716298198715678139084995,
0.58364716298198715678139084995, 2.08573337518142273594511285902, 2.84698544346114256596719804471, 3.43992235223035630666817469070, 4.55650226276854385696390888979, 5.05230414477415383945981497923, 5.44722087764392814401734091744, 5.94094495435168887474430200565, 7.00270559397515197426920825528, 7.59123761733401727333375368453
