| L(s) = 1 | − 0.353·2-s − 2.88·3-s − 1.87·4-s − 4.41·5-s + 1.02·6-s − 3.52·7-s + 1.36·8-s + 5.35·9-s + 1.56·10-s + 11-s + 5.41·12-s − 1.28·13-s + 1.24·14-s + 12.7·15-s + 3.26·16-s + 17-s − 1.89·18-s − 6.94·19-s + 8.28·20-s + 10.1·21-s − 0.353·22-s − 3.98·23-s − 3.95·24-s + 14.5·25-s + 0.453·26-s − 6.79·27-s + 6.61·28-s + ⋯ |
| L(s) = 1 | − 0.249·2-s − 1.66·3-s − 0.937·4-s − 1.97·5-s + 0.416·6-s − 1.33·7-s + 0.483·8-s + 1.78·9-s + 0.493·10-s + 0.301·11-s + 1.56·12-s − 0.355·13-s + 0.332·14-s + 3.29·15-s + 0.816·16-s + 0.242·17-s − 0.445·18-s − 1.59·19-s + 1.85·20-s + 2.22·21-s − 0.0753·22-s − 0.830·23-s − 0.807·24-s + 2.90·25-s + 0.0888·26-s − 1.30·27-s + 1.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03846697795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03846697795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.353T + 2T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 + 4.41T + 5T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 19 | \( 1 + 6.94T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 4.05T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 0.367T + 41T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 1.31T + 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 2.21T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 2.25T + 73T^{2} \) |
| 79 | \( 1 - 8.95T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 - 4.36T + 97T^{2} \) |
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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
−7.71100476640874658687749210926, −7.10657489471365619033647791762, −6.43028297001072236839797397022, −5.86324321788653508914122972387, −4.83702411355448616354357992333, −4.27096686213420833697947655979, −3.94951140448758341696021456629, −2.94401349101978415871139837172, −1.02967028054419721926604611992, −0.14282647663376329359965932210,
0.14282647663376329359965932210, 1.02967028054419721926604611992, 2.94401349101978415871139837172, 3.94951140448758341696021456629, 4.27096686213420833697947655979, 4.83702411355448616354357992333, 5.86324321788653508914122972387, 6.43028297001072236839797397022, 7.10657489471365619033647791762, 7.71100476640874658687749210926
