| L(s) = 1 | − 2.09·2-s + 2.34·3-s + 2.40·4-s − 0.402·5-s − 4.92·6-s + 0.678·7-s − 0.843·8-s + 2.50·9-s + 0.843·10-s + 0.745·11-s + 5.63·12-s + 6.09·13-s − 1.42·14-s − 0.943·15-s − 3.03·16-s − 6.59·17-s − 5.24·18-s + 0.827·19-s − 0.965·20-s + 1.59·21-s − 1.56·22-s − 7.71·23-s − 1.97·24-s − 4.83·25-s − 12.7·26-s − 1.17·27-s + 1.63·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.35·3-s + 1.20·4-s − 0.179·5-s − 2.00·6-s + 0.256·7-s − 0.298·8-s + 0.833·9-s + 0.266·10-s + 0.224·11-s + 1.62·12-s + 1.68·13-s − 0.380·14-s − 0.243·15-s − 0.758·16-s − 1.59·17-s − 1.23·18-s + 0.189·19-s − 0.216·20-s + 0.347·21-s − 0.333·22-s − 1.60·23-s − 0.403·24-s − 0.967·25-s − 2.50·26-s − 0.225·27-s + 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7364228303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7364228303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 89 | \( 1 - T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 + 0.402T + 5T^{2} \) |
| 7 | \( 1 - 0.678T + 7T^{2} \) |
| 11 | \( 1 - 0.745T + 11T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 + 6.59T + 17T^{2} \) |
| 19 | \( 1 - 0.827T + 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 - 0.400T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 - 3.45T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 0.597T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−14.04334995036929826804629448041, −13.46134444490306456603544396266, −11.61908377266560458486313569278, −10.55512493808139176898983968487, −9.380097245208422507260802364394, −8.508260957313873413365094327982, −8.050516926738481523314434039074, −6.58266311964925475644416378412, −3.91926158936289874458722067095, −1.98034343477875218669591159961,
1.98034343477875218669591159961, 3.91926158936289874458722067095, 6.58266311964925475644416378412, 8.050516926738481523314434039074, 8.508260957313873413365094327982, 9.380097245208422507260802364394, 10.55512493808139176898983968487, 11.61908377266560458486313569278, 13.46134444490306456603544396266, 14.04334995036929826804629448041
