| L(s) = 1 | − 0.628·2-s − 2.03·3-s − 1.60·4-s + 1.28·6-s + 0.952·7-s + 2.26·8-s + 1.14·9-s + 0.264·11-s + 3.26·12-s + 4.33·13-s − 0.598·14-s + 1.78·16-s − 5.23·17-s − 0.720·18-s + 5.12·19-s − 1.93·21-s − 0.166·22-s − 0.0255·23-s − 4.61·24-s − 2.72·26-s + 3.77·27-s − 1.52·28-s + 0.821·29-s − 2.30·31-s − 5.65·32-s − 0.538·33-s + 3.29·34-s + ⋯ |
| L(s) = 1 | − 0.444·2-s − 1.17·3-s − 0.802·4-s + 0.522·6-s + 0.359·7-s + 0.801·8-s + 0.381·9-s + 0.0797·11-s + 0.943·12-s + 1.20·13-s − 0.159·14-s + 0.446·16-s − 1.27·17-s − 0.169·18-s + 1.17·19-s − 0.423·21-s − 0.0354·22-s − 0.00532·23-s − 0.942·24-s − 0.534·26-s + 0.726·27-s − 0.288·28-s + 0.152·29-s − 0.413·31-s − 0.999·32-s − 0.0937·33-s + 0.564·34-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
| \(L(1)\) |
\(\approx\) |
\(0.8393699374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8393699374\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.628T + 2T^{2} \) |
| 3 | \( 1 + 2.03T + 3T^{2} \) |
| 7 | \( 1 - 0.952T + 7T^{2} \) |
| 11 | \( 1 - 0.264T + 11T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 0.0255T + 23T^{2} \) |
| 29 | \( 1 - 0.821T + 29T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 6.04T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 + 3.68T + 61T^{2} \) |
| 67 | \( 1 + 7.24T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 - 7.27T + 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 + 5.32T + 83T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.065926915717403122520293621169, −7.47402255674015206213552805841, −6.54842109405013980343924641297, −5.83499678799235736895000753432, −5.34951953461543510351944727338, −4.42509241688207411934714020334, −4.02313768698037415296555276432, −2.72237710082002225282916365138, −1.32795562398448902100445063255, −0.63470216782241702163808942781,
0.63470216782241702163808942781, 1.32795562398448902100445063255, 2.72237710082002225282916365138, 4.02313768698037415296555276432, 4.42509241688207411934714020334, 5.34951953461543510351944727338, 5.83499678799235736895000753432, 6.54842109405013980343924641297, 7.47402255674015206213552805841, 8.065926915717403122520293621169
