L(s) = 1 | + 1.83·3-s − 1.83·7-s + 0.364·9-s − 0.834·11-s + 2.19·13-s − 2.56·17-s + 19-s − 3.36·21-s + 0.635·23-s − 4.83·27-s − 9.62·29-s + 6.59·31-s − 1.53·33-s + 5.23·37-s + 4.03·39-s + 4.43·41-s + 7.06·43-s − 9.86·47-s − 3.63·49-s − 4.70·51-s + 0.668·53-s + 1.83·57-s − 0.397·59-s + 2.26·61-s − 0.668·63-s − 2.43·67-s + 1.16·69-s + ⋯ |
L(s) = 1 | + 1.05·3-s − 0.693·7-s + 0.121·9-s − 0.251·11-s + 0.609·13-s − 0.621·17-s + 0.229·19-s − 0.734·21-s + 0.132·23-s − 0.930·27-s − 1.78·29-s + 1.18·31-s − 0.266·33-s + 0.860·37-s + 0.645·39-s + 0.691·41-s + 1.07·43-s − 1.43·47-s − 0.519·49-s − 0.658·51-s + 0.0918·53-s + 0.242·57-s − 0.0517·59-s + 0.289·61-s − 0.0842·63-s − 0.297·67-s + 0.140·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + 0.834T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 23 | \( 1 - 0.635T + 23T^{2} \) |
| 29 | \( 1 + 9.62T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 7.06T + 43T^{2} \) |
| 47 | \( 1 + 9.86T + 47T^{2} \) |
| 53 | \( 1 - 0.668T + 53T^{2} \) |
| 59 | \( 1 + 0.397T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 + 9.49T + 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.65200968235169228383929825111, −6.95154335933671369230687337081, −6.12291265208377251563413952278, −5.58669393218499469071999644897, −4.45792477613532140399350236912, −3.80416989806743059830702342128, −3.01700655284271748452201322061, −2.49793557343575314064647794033, −1.44018648196099238543104718524, 0,
1.44018648196099238543104718524, 2.49793557343575314064647794033, 3.01700655284271748452201322061, 3.80416989806743059830702342128, 4.45792477613532140399350236912, 5.58669393218499469071999644897, 6.12291265208377251563413952278, 6.95154335933671369230687337081, 7.65200968235169228383929825111