L(s) = 1 | + 2-s + 2.56·3-s + 4-s + 2·5-s + 2.56·6-s + 0.561·7-s + 8-s + 3.56·9-s + 2·10-s + 2.56·12-s − 0.561·13-s + 0.561·14-s + 5.12·15-s + 16-s + 0.561·17-s + 3.56·18-s + 19-s + 2·20-s + 1.43·21-s + 1.43·23-s + 2.56·24-s − 25-s − 0.561·26-s + 1.43·27-s + 0.561·28-s + 5.68·29-s + 5.12·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.47·3-s + 0.5·4-s + 0.894·5-s + 1.04·6-s + 0.212·7-s + 0.353·8-s + 1.18·9-s + 0.632·10-s + 0.739·12-s − 0.155·13-s + 0.150·14-s + 1.32·15-s + 0.250·16-s + 0.136·17-s + 0.839·18-s + 0.229·19-s + 0.447·20-s + 0.313·21-s + 0.299·23-s + 0.522·24-s − 0.200·25-s − 0.110·26-s + 0.276·27-s + 0.106·28-s + 1.05·29-s + 0.935·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.595583724\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.595583724\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 - 0.561T + 17T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 9.68T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 0.876T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−8.354379142627072384789793367177, −7.60694106066221753151363001908, −6.92851520915621082450923490537, −6.08598271570283301646587712261, −5.30865246335906277681277042760, −4.49550810085868735906681271510, −3.63032619239774462019841399777, −2.85846560900900947603954021878, −2.23629020885531582096547969005, −1.37839871735081611330517311973,
1.37839871735081611330517311973, 2.23629020885531582096547969005, 2.85846560900900947603954021878, 3.63032619239774462019841399777, 4.49550810085868735906681271510, 5.30865246335906277681277042760, 6.08598271570283301646587712261, 6.92851520915621082450923490537, 7.60694106066221753151363001908, 8.354379142627072384789793367177