| L(s) = 1 | − 2.36·2-s − 0.380·3-s + 3.59·4-s + 0.898·6-s − 4.83·7-s − 3.76·8-s − 2.85·9-s + 4.47·11-s − 1.36·12-s + 5.34·13-s + 11.4·14-s + 1.72·16-s − 5.51·17-s + 6.75·18-s + 3.77·19-s + 1.83·21-s − 10.5·22-s + 3.44·23-s + 1.43·24-s − 12.6·26-s + 2.22·27-s − 17.3·28-s − 4.63·29-s − 0.722·31-s + 3.46·32-s − 1.70·33-s + 13.0·34-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 0.219·3-s + 1.79·4-s + 0.366·6-s − 1.82·7-s − 1.33·8-s − 0.951·9-s + 1.34·11-s − 0.394·12-s + 1.48·13-s + 3.05·14-s + 0.430·16-s − 1.33·17-s + 1.59·18-s + 0.866·19-s + 0.400·21-s − 2.25·22-s + 0.718·23-s + 0.292·24-s − 2.48·26-s + 0.428·27-s − 3.27·28-s − 0.861·29-s − 0.129·31-s + 0.612·32-s − 0.296·33-s + 2.23·34-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
| \(L(1)\) |
\(\approx\) |
\(0.4682472056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4682472056\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 + 0.380T + 3T^{2} \) |
| 7 | \( 1 + 4.83T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 - 3.44T + 23T^{2} \) |
| 29 | \( 1 + 4.63T + 29T^{2} \) |
| 31 | \( 1 + 0.722T + 31T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 - 6.18T + 41T^{2} \) |
| 43 | \( 1 + 0.997T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 - 4.28T + 53T^{2} \) |
| 59 | \( 1 + 6.23T + 59T^{2} \) |
| 61 | \( 1 - 0.376T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 + 8.06T + 71T^{2} \) |
| 73 | \( 1 - 6.15T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−8.480849418742294670559172833018, −7.24843415929792948058411459110, −6.85953157206810202622948985354, −6.14729106705806706984905776885, −5.80552394012779935201019668429, −4.18811708280048137287805079974, −3.37424034263014339481068848824, −2.65547289459777635983694732447, −1.42083697385187758564333588377, −0.49582481348407633491882487172,
0.49582481348407633491882487172, 1.42083697385187758564333588377, 2.65547289459777635983694732447, 3.37424034263014339481068848824, 4.18811708280048137287805079974, 5.80552394012779935201019668429, 6.14729106705806706984905776885, 6.85953157206810202622948985354, 7.24843415929792948058411459110, 8.480849418742294670559172833018
