| L(s) = 1 | − 2·2-s + 2·4-s + 4·7-s + 11-s − 5·13-s − 8·14-s − 4·16-s + 6·17-s − 7·19-s − 2·22-s + 23-s + 10·26-s + 8·28-s − 29-s + 8·31-s + 8·32-s − 12·34-s − 9·37-s + 14·38-s − 12·41-s + 43-s + 2·44-s − 2·46-s − 10·47-s + 9·49-s − 10·52-s − 8·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.51·7-s + 0.301·11-s − 1.38·13-s − 2.13·14-s − 16-s + 1.45·17-s − 1.60·19-s − 0.426·22-s + 0.208·23-s + 1.96·26-s + 1.51·28-s − 0.185·29-s + 1.43·31-s + 1.41·32-s − 2.05·34-s − 1.47·37-s + 2.27·38-s − 1.87·41-s + 0.152·43-s + 0.301·44-s − 0.294·46-s − 1.45·47-s + 9/7·49-s − 1.38·52-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7991727385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7991727385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−13.40165523874289, −12.63360845262876, −12.25392980760811, −11.67275388681985, −11.46675526558474, −10.74648772774706, −10.30909720221413, −10.06931037841142, −9.528397037023705, −8.847495425148974, −8.531933655615355, −7.958701440805847, −7.817520639830849, −7.189143303425560, −6.657388676419359, −6.149685982315707, −5.194226371636382, −4.817142250798187, −4.579003920325460, −3.635251013471967, −2.946313399741301, −2.078129206689467, −1.779871762604597, −1.214824456114703, −0.3477173495324318,
0.3477173495324318, 1.214824456114703, 1.779871762604597, 2.078129206689467, 2.946313399741301, 3.635251013471967, 4.579003920325460, 4.817142250798187, 5.194226371636382, 6.149685982315707, 6.657388676419359, 7.189143303425560, 7.817520639830849, 7.958701440805847, 8.531933655615355, 8.847495425148974, 9.528397037023705, 10.06931037841142, 10.30909720221413, 10.74648772774706, 11.46675526558474, 11.67275388681985, 12.25392980760811, 12.63360845262876, 13.40165523874289
