L(s) = 1 | + 1.52·2-s − 3-s + 0.318·4-s − 1.52·6-s + 0.990·7-s − 2.56·8-s + 9-s + 5.97·11-s − 0.318·12-s − 4.02·13-s + 1.50·14-s − 4.53·16-s − 0.476·17-s + 1.52·18-s + 2.82·19-s − 0.990·21-s + 9.09·22-s + 1.74·23-s + 2.56·24-s − 6.13·26-s − 27-s + 0.315·28-s − 1.41·29-s + 8.76·31-s − 1.78·32-s − 5.97·33-s − 0.725·34-s + ⋯ |
L(s) = 1 | + 1.07·2-s − 0.577·3-s + 0.159·4-s − 0.621·6-s + 0.374·7-s − 0.905·8-s + 0.333·9-s + 1.80·11-s − 0.0918·12-s − 1.11·13-s + 0.403·14-s − 1.13·16-s − 0.115·17-s + 0.358·18-s + 0.647·19-s − 0.216·21-s + 1.93·22-s + 0.364·23-s + 0.522·24-s − 1.20·26-s − 0.192·27-s + 0.0596·28-s − 0.263·29-s + 1.57·31-s − 0.315·32-s − 1.03·33-s − 0.124·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.426505154\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.426505154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 1.52T + 2T^{2} \) |
| 7 | \( 1 - 0.990T + 7T^{2} \) |
| 11 | \( 1 - 5.97T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 17 | \( 1 + 0.476T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8.76T + 31T^{2} \) |
| 37 | \( 1 - 8.06T + 37T^{2} \) |
| 41 | \( 1 + 5.50T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 - 9.62T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 5.43T + 71T^{2} \) |
| 73 | \( 1 - 4.08T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−9.396901091586162345329127990893, −8.498551143249724369506608682064, −7.36000322665621995123435200875, −6.59160589701814816944002661779, −5.93365131568117995402403810238, −5.00349562706998748510937248972, −4.43692901744711452359530361763, −3.65416429453972069237429275972, −2.47681203859761797379330953726, −0.965663755401262549845803972703,
0.965663755401262549845803972703, 2.47681203859761797379330953726, 3.65416429453972069237429275972, 4.43692901744711452359530361763, 5.00349562706998748510937248972, 5.93365131568117995402403810238, 6.59160589701814816944002661779, 7.36000322665621995123435200875, 8.498551143249724369506608682064, 9.396901091586162345329127990893