L(s) = 1 | − 3-s + 3·5-s − 2·9-s + 3·11-s + 2·13-s − 3·15-s + 3·17-s + 19-s − 3·23-s + 4·25-s + 5·27-s − 6·29-s + 7·31-s − 3·33-s − 37-s − 2·39-s + 6·41-s + 4·43-s − 6·45-s + 9·47-s − 3·51-s + 3·53-s + 9·55-s − 57-s − 9·59-s − 61-s + 6·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.904·11-s + 0.554·13-s − 0.774·15-s + 0.727·17-s + 0.229·19-s − 0.625·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s + 1.25·31-s − 0.522·33-s − 0.164·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.894·45-s + 1.31·47-s − 0.420·51-s + 0.412·53-s + 1.21·55-s − 0.132·57-s − 1.17·59-s − 0.128·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.648272273\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.648272273\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.28821554628842933399761045452, −9.465538249257530723796427509649, −8.832835367079797434619327214039, −7.68892345677099146953256519867, −6.40175986647281313923469473601, −5.96957053958209996558586270063, −5.22421181432809928276365314617, −3.86935901775791325818436521316, −2.51980819633185345991452121168, −1.19485794360651508706892872923,
1.19485794360651508706892872923, 2.51980819633185345991452121168, 3.86935901775791325818436521316, 5.22421181432809928276365314617, 5.96957053958209996558586270063, 6.40175986647281313923469473601, 7.68892345677099146953256519867, 8.832835367079797434619327214039, 9.465538249257530723796427509649, 10.28821554628842933399761045452