| L(s) = 1 | − 3-s + 9-s − 4·11-s − 13-s + 6·17-s − 4·19-s + 8·23-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 10·37-s + 39-s − 6·41-s + 4·43-s − 7·49-s − 6·51-s + 10·53-s + 4·57-s − 4·59-s − 2·61-s − 12·67-s − 8·69-s − 16·71-s − 2·73-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 1.64·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s − 49-s − 0.840·51-s + 1.37·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s − 1.46·67-s − 0.963·69-s − 1.89·71-s − 0.234·73-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584096006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584096006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.09064519991964, −15.42497182974750, −14.93740870610681, −14.50853289193865, −13.57535293497016, −13.20095375062276, −12.66493233546466, −12.01618251217708, −11.64671534050213, −10.73169669695177, −10.44885334051563, −9.948949993387420, −9.206176560275258, −8.443772401254047, −7.857287469914161, −7.373126163827036, −6.549633078570919, −6.041104234665362, −5.249821538335546, −4.828736491820654, −4.149278790780585, −2.943457407954193, −2.721788876585816, −1.404615511721431, −0.6006427229869228,
0.6006427229869228, 1.404615511721431, 2.721788876585816, 2.943457407954193, 4.149278790780585, 4.828736491820654, 5.249821538335546, 6.041104234665362, 6.549633078570919, 7.373126163827036, 7.857287469914161, 8.443772401254047, 9.206176560275258, 9.948949993387420, 10.44885334051563, 10.73169669695177, 11.64671534050213, 12.01618251217708, 12.66493233546466, 13.20095375062276, 13.57535293497016, 14.50853289193865, 14.93740870610681, 15.42497182974750, 16.09064519991964
