| L(s) = 1 | + 0.343·3-s + 1.47·5-s + 1.11·7-s − 2.88·9-s − 0.662·11-s + 1.41·13-s + 0.506·15-s − 2.59·17-s + 0.818·19-s + 0.383·21-s − 6.34·23-s − 2.82·25-s − 2.01·27-s + 8.03·29-s + 0.503·31-s − 0.227·33-s + 1.64·35-s − 2.94·37-s + 0.485·39-s − 6.86·41-s − 1.07·43-s − 4.25·45-s + 2.38·47-s − 5.75·49-s − 0.891·51-s − 5.12·53-s − 0.976·55-s + ⋯ |
| L(s) = 1 | + 0.198·3-s + 0.659·5-s + 0.422·7-s − 0.960·9-s − 0.199·11-s + 0.392·13-s + 0.130·15-s − 0.629·17-s + 0.187·19-s + 0.0837·21-s − 1.32·23-s − 0.564·25-s − 0.388·27-s + 1.49·29-s + 0.0903·31-s − 0.0395·33-s + 0.278·35-s − 0.484·37-s + 0.0778·39-s − 1.07·41-s − 0.163·43-s − 0.633·45-s + 0.348·47-s − 0.821·49-s − 0.124·51-s − 0.703·53-s − 0.131·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.343T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 + 0.662T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 0.818T + 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 - 0.503T + 31T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 1.40T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 + 4.77T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 + 2.84T + 73T^{2} \) |
| 79 | \( 1 + 3.34T + 79T^{2} \) |
| 83 | \( 1 + 3.93T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 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
−8.052186176078761973478594520303, −6.90896644617807401667230338034, −6.23413161624440432611366756728, −5.64292519266268748097301985615, −4.91773224446196616805222870641, −4.05004913526337368220172227334, −3.10141313767911649031869204277, −2.31083941837928631674702735128, −1.50676359460433654690388578196, 0,
1.50676359460433654690388578196, 2.31083941837928631674702735128, 3.10141313767911649031869204277, 4.05004913526337368220172227334, 4.91773224446196616805222870641, 5.64292519266268748097301985615, 6.23413161624440432611366756728, 6.90896644617807401667230338034, 8.052186176078761973478594520303
