| L(s) = 1 | − 0.128·2-s + 3-s − 1.98·4-s + 5-s − 0.128·6-s − 7-s + 0.513·8-s + 9-s − 0.128·10-s − 6.08·11-s − 1.98·12-s − 6.03·13-s + 0.128·14-s + 15-s + 3.90·16-s − 5.30·17-s − 0.128·18-s + 5.72·19-s − 1.98·20-s − 21-s + 0.783·22-s + 23-s + 0.513·24-s + 25-s + 0.777·26-s + 27-s + 1.98·28-s + ⋯ |
| L(s) = 1 | − 0.0910·2-s + 0.577·3-s − 0.991·4-s + 0.447·5-s − 0.0525·6-s − 0.377·7-s + 0.181·8-s + 0.333·9-s − 0.0407·10-s − 1.83·11-s − 0.572·12-s − 1.67·13-s + 0.0344·14-s + 0.258·15-s + 0.975·16-s − 1.28·17-s − 0.0303·18-s + 1.31·19-s − 0.443·20-s − 0.218·21-s + 0.167·22-s + 0.208·23-s + 0.104·24-s + 0.200·25-s + 0.152·26-s + 0.192·27-s + 0.374·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.277531822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277531822\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.128T + 2T^{2} \) |
| 11 | \( 1 + 6.08T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 8.72T + 31T^{2} \) |
| 37 | \( 1 - 4.50T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 + 6.64T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 8.75T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.94T + 73T^{2} \) |
| 79 | \( 1 + 0.792T + 79T^{2} \) |
| 83 | \( 1 - 0.255T + 83T^{2} \) |
| 89 | \( 1 + 8.58T + 89T^{2} \) |
| 97 | \( 1 + 2.05T + 97T^{2} \) |
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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
−9.045416667222296997361284094116, −8.165758519841696012963122101277, −7.65969168977692564566495008384, −6.79461608571490305527425993555, −5.61399898657053032617697995313, −4.88478493576816687343230739526, −4.36326572786335100310731626415, −2.76671509789082644381915201551, −2.61119456847869464426943861268, −0.69411003110629324672491394078,
0.69411003110629324672491394078, 2.61119456847869464426943861268, 2.76671509789082644381915201551, 4.36326572786335100310731626415, 4.88478493576816687343230739526, 5.61399898657053032617697995313, 6.79461608571490305527425993555, 7.65969168977692564566495008384, 8.165758519841696012963122101277, 9.045416667222296997361284094116
