L(s) = 1 | + 2-s − 4-s − 5-s − 1.73·7-s − 3·8-s − 10-s − 2.46·11-s − 13-s − 1.73·14-s − 16-s + 4·17-s − 3.73·19-s + 20-s − 2.46·22-s − 2.53·23-s + 25-s − 26-s + 1.73·28-s − 9.19·29-s − 3.46·31-s + 5·32-s + 4·34-s + 1.73·35-s + 5.46·37-s − 3.73·38-s + 3·40-s + 5.46·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.5·4-s − 0.447·5-s − 0.654·7-s − 1.06·8-s − 0.316·10-s − 0.742·11-s − 0.277·13-s − 0.462·14-s − 0.250·16-s + 0.970·17-s − 0.856·19-s + 0.223·20-s − 0.525·22-s − 0.528·23-s + 0.200·25-s − 0.196·26-s + 0.327·28-s − 1.70·29-s − 0.622·31-s + 0.883·32-s + 0.685·34-s + 0.292·35-s + 0.898·37-s − 0.605·38-s + 0.474·40-s + 0.853·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9865042434\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9865042434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 7.19T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.0717T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 7.53T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 3.92T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.042110708944470752587020677634, −7.60706330206946760629702637749, −6.59263847449316555743168579606, −5.85784862887490951598400528117, −5.28717938481197015989314425102, −4.46736991500953031110997026301, −3.71460671649808146872757129381, −3.16049245440554645680822853936, −2.13569751346220812836527763170, −0.45910629918847979306160362458,
0.45910629918847979306160362458, 2.13569751346220812836527763170, 3.16049245440554645680822853936, 3.71460671649808146872757129381, 4.46736991500953031110997026301, 5.28717938481197015989314425102, 5.85784862887490951598400528117, 6.59263847449316555743168579606, 7.60706330206946760629702637749, 8.042110708944470752587020677634