| L(s) = 1 | − 2.41·3-s − 5-s + 2.82·9-s + 4.82·11-s − 2·13-s + 2.41·15-s − 3.65·17-s + 5.65·19-s + 8.41·23-s + 25-s + 0.414·27-s − 2.17·29-s − 4.82·31-s − 11.6·33-s − 5.65·37-s + 4.82·39-s − 0.171·41-s − 12.8·43-s − 2.82·45-s + 0.343·47-s + 8.82·51-s + 5.65·53-s − 4.82·55-s − 13.6·57-s − 4·59-s + 4.65·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 0.447·5-s + 0.942·9-s + 1.45·11-s − 0.554·13-s + 0.623·15-s − 0.886·17-s + 1.29·19-s + 1.75·23-s + 0.200·25-s + 0.0797·27-s − 0.403·29-s − 0.867·31-s − 2.02·33-s − 0.929·37-s + 0.773·39-s − 0.0267·41-s − 1.96·43-s − 0.421·45-s + 0.0500·47-s + 1.23·51-s + 0.777·53-s − 0.651·55-s − 1.80·57-s − 0.520·59-s + 0.596·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9419866289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9419866289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8.41T + 23T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 + 0.171T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 4.65T + 61T^{2} \) |
| 67 | \( 1 + 6.89T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.606639025731844223129765003900, −7.38067759437325940149333477478, −6.89598350563365774795898987171, −6.38859648762189023262285128765, −5.28601376554767630912251357432, −4.98155971055383183762337452585, −3.98240485332028859171618924374, −3.15033913867567121032627451940, −1.64124097874298841744850039211, −0.62526435522721393061037845624,
0.62526435522721393061037845624, 1.64124097874298841744850039211, 3.15033913867567121032627451940, 3.98240485332028859171618924374, 4.98155971055383183762337452585, 5.28601376554767630912251357432, 6.38859648762189023262285128765, 6.89598350563365774795898987171, 7.38067759437325940149333477478, 8.606639025731844223129765003900
