| L(s) = 1 | + 2.48·2-s + 3-s + 4.16·4-s + 5-s + 2.48·6-s + 4.94·7-s + 5.36·8-s + 9-s + 2.48·10-s + 0.391·11-s + 4.16·12-s + 3.57·13-s + 12.2·14-s + 15-s + 4.99·16-s − 6.91·17-s + 2.48·18-s − 3.84·19-s + 4.16·20-s + 4.94·21-s + 0.971·22-s − 3.69·23-s + 5.36·24-s + 25-s + 8.87·26-s + 27-s + 20.5·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.577·3-s + 2.08·4-s + 0.447·5-s + 1.01·6-s + 1.87·7-s + 1.89·8-s + 0.333·9-s + 0.784·10-s + 0.118·11-s + 1.20·12-s + 0.991·13-s + 3.28·14-s + 0.258·15-s + 1.24·16-s − 1.67·17-s + 0.585·18-s − 0.881·19-s + 0.930·20-s + 1.07·21-s + 0.207·22-s − 0.770·23-s + 1.09·24-s + 0.200·25-s + 1.74·26-s + 0.192·27-s + 3.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.12105583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.12105583\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 - 0.391T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 6.91T + 17T^{2} \) |
| 19 | \( 1 + 3.84T + 19T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 - 0.710T + 29T^{2} \) |
| 31 | \( 1 - 0.293T + 31T^{2} \) |
| 37 | \( 1 - 4.13T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 + 7.36T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 1.94T + 71T^{2} \) |
| 73 | \( 1 - 5.92T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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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
−8.007107112972658682292803624622, −7.20995206849539050718421433180, −6.32215230017559143580218196742, −5.92539030671780303145481536974, −4.91258589447344327944612932639, −4.41327667254206989577513733768, −4.00325043304708249073105435367, −2.83331918009562797118797235947, −2.05591628900252036242137201719, −1.55556227210558991386003889820,
1.55556227210558991386003889820, 2.05591628900252036242137201719, 2.83331918009562797118797235947, 4.00325043304708249073105435367, 4.41327667254206989577513733768, 4.91258589447344327944612932639, 5.92539030671780303145481536974, 6.32215230017559143580218196742, 7.20995206849539050718421433180, 8.007107112972658682292803624622
