| L(s) = 1 | + 1.56·2-s + 0.438·4-s + 4.56·7-s − 2.43·8-s + 2.56·11-s + 13-s + 7.12·14-s − 4.68·16-s + 2.56·17-s + 3.12·19-s + 4·22-s − 6.56·23-s + 1.56·26-s + 1.99·28-s + 1.12·29-s + 6·31-s − 2.43·32-s + 4·34-s − 1.68·37-s + 4.87·38-s − 0.561·41-s + 5.12·43-s + 1.12·44-s − 10.2·46-s − 2.87·47-s + 13.8·49-s + 0.438·52-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.219·4-s + 1.72·7-s − 0.862·8-s + 0.772·11-s + 0.277·13-s + 1.90·14-s − 1.17·16-s + 0.621·17-s + 0.716·19-s + 0.852·22-s − 1.36·23-s + 0.306·26-s + 0.377·28-s + 0.208·29-s + 1.07·31-s − 0.431·32-s + 0.685·34-s − 0.276·37-s + 0.791·38-s − 0.0876·41-s + 0.781·43-s + 0.169·44-s − 1.51·46-s − 0.419·47-s + 1.97·49-s + 0.0608·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.860779107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.860779107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + 0.561T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 2.87T + 47T^{2} \) |
| 53 | \( 1 + 7.68T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 7.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 1.68T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.533768314648039439477523118229, −8.099892039610362740320999595846, −7.17780812217765016168627638595, −6.16745293178126316377869046072, −5.53968201568936277872707289203, −4.77725188200614312432605684970, −4.19376000668924508543344728870, −3.39478808647222406723076638630, −2.20456632767996594601131477438, −1.12380167435003520088277888790,
1.12380167435003520088277888790, 2.20456632767996594601131477438, 3.39478808647222406723076638630, 4.19376000668924508543344728870, 4.77725188200614312432605684970, 5.53968201568936277872707289203, 6.16745293178126316377869046072, 7.17780812217765016168627638595, 8.099892039610362740320999595846, 8.533768314648039439477523118229
