| L(s) = 1 | − 0.148·3-s − 0.337·5-s − 7-s − 2.97·9-s − 11-s − 13-s + 0.0503·15-s − 4.57·17-s − 7.18·19-s + 0.148·21-s − 1.19·23-s − 4.88·25-s + 0.890·27-s + 5.73·29-s + 2.83·31-s + 0.148·33-s + 0.337·35-s + 4.01·37-s + 0.148·39-s − 5.31·41-s − 5.67·43-s + 1.00·45-s − 0.233·47-s + 49-s + 0.681·51-s + 1.32·53-s + 0.337·55-s + ⋯ |
| L(s) = 1 | − 0.0859·3-s − 0.151·5-s − 0.377·7-s − 0.992·9-s − 0.301·11-s − 0.277·13-s + 0.0129·15-s − 1.11·17-s − 1.64·19-s + 0.0324·21-s − 0.250·23-s − 0.977·25-s + 0.171·27-s + 1.06·29-s + 0.509·31-s + 0.0259·33-s + 0.0571·35-s + 0.660·37-s + 0.0238·39-s − 0.830·41-s − 0.865·43-s + 0.150·45-s − 0.0340·47-s + 0.142·49-s + 0.0954·51-s + 0.181·53-s + 0.0455·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6498306692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6498306692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.148T + 3T^{2} \) |
| 5 | \( 1 + 0.337T + 5T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 + 7.18T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 + 0.233T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 + 2.57T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 9.26T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 + 7.82T + 73T^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 0.913T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.998679623647548088089507254438, −7.04819170590907664501069922407, −6.30927297883832698197985753373, −5.99995222776714511781701615583, −4.91370798822351545048310922558, −4.39556505530465273808642441672, −3.46710430157416162001076078986, −2.61544995376478860245528665299, −1.97434061277933007173589502984, −0.37302375855318762267446774610,
0.37302375855318762267446774610, 1.97434061277933007173589502984, 2.61544995376478860245528665299, 3.46710430157416162001076078986, 4.39556505530465273808642441672, 4.91370798822351545048310922558, 5.99995222776714511781701615583, 6.30927297883832698197985753373, 7.04819170590907664501069922407, 7.998679623647548088089507254438
