| L(s) = 1 | − 2.19·2-s + 2.83·3-s + 2.83·4-s − 6.23·6-s − 1.83·8-s + 5.03·9-s − 2.56·11-s + 8.03·12-s − 0.563·13-s − 1.63·16-s + 5.19·17-s − 11.0·18-s + 0.469·19-s + 5.63·22-s + 4.03·23-s − 5.19·24-s + 1.23·26-s + 5.76·27-s + 2.86·29-s − 3.86·31-s + 7.26·32-s − 7.26·33-s − 11.4·34-s + 14.2·36-s + 2.23·37-s − 1.03·38-s − 1.59·39-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.63·3-s + 1.41·4-s − 2.54·6-s − 0.648·8-s + 1.67·9-s − 0.772·11-s + 2.31·12-s − 0.156·13-s − 0.408·16-s + 1.26·17-s − 2.60·18-s + 0.107·19-s + 1.20·22-s + 0.840·23-s − 1.06·24-s + 0.242·26-s + 1.10·27-s + 0.532·29-s − 0.694·31-s + 1.28·32-s − 1.26·33-s − 1.96·34-s + 2.37·36-s + 0.366·37-s − 0.167·38-s − 0.255·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402396494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402396494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 - 2.83T + 3T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 0.563T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 0.469T + 19T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.36T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 3.96T + 59T^{2} \) |
| 61 | \( 1 + 3.86T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 + 8.73T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 - 3.13T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.419355467998863216736163975892, −8.958047618049415763860679975590, −8.232807950980087931427704485090, −7.52100132833698935940463949088, −7.21456982615170003024345492212, −5.67786621964899772108255052479, −4.27805649680129147079881681361, −3.03411744596931956768956453859, −2.33721041997254496441563645696, −1.08736660593658112207671698534,
1.08736660593658112207671698534, 2.33721041997254496441563645696, 3.03411744596931956768956453859, 4.27805649680129147079881681361, 5.67786621964899772108255052479, 7.21456982615170003024345492212, 7.52100132833698935940463949088, 8.232807950980087931427704485090, 8.958047618049415763860679975590, 9.419355467998863216736163975892
