L(s) = 1 | + 2-s − 3.24·3-s + 4-s + 5-s − 3.24·6-s + 1.32·7-s + 8-s + 7.50·9-s + 10-s − 4.26·11-s − 3.24·12-s − 5.76·13-s + 1.32·14-s − 3.24·15-s + 16-s − 5.87·17-s + 7.50·18-s − 1.43·19-s + 20-s − 4.28·21-s − 4.26·22-s + 9.35·23-s − 3.24·24-s + 25-s − 5.76·26-s − 14.6·27-s + 1.32·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.87·3-s + 0.5·4-s + 0.447·5-s − 1.32·6-s + 0.499·7-s + 0.353·8-s + 2.50·9-s + 0.316·10-s − 1.28·11-s − 0.935·12-s − 1.59·13-s + 0.353·14-s − 0.837·15-s + 0.250·16-s − 1.42·17-s + 1.76·18-s − 0.329·19-s + 0.223·20-s − 0.935·21-s − 0.908·22-s + 1.95·23-s − 0.661·24-s + 0.200·25-s − 1.12·26-s − 2.81·27-s + 0.249·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.329921585\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.329921585\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 + 5.76T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 23 | \( 1 - 9.35T + 23T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 9.02T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 - 3.84T + 53T^{2} \) |
| 59 | \( 1 + 2.45T + 59T^{2} \) |
| 61 | \( 1 - 2.73T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 8.65T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 - 1.06T + 97T^{2} \) |
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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
−8.211123526792956143008172075081, −7.30506309695120754539725350015, −6.76592262620591793240938009722, −6.15140925908559062973423211653, −5.22664482048305214795586640318, −4.79715310947635289487596726359, −4.56991281471844814515420537170, −2.84248625946737637259641978911, −2.00076041155585510755943376040, −0.63583804852645936140515090917,
0.63583804852645936140515090917, 2.00076041155585510755943376040, 2.84248625946737637259641978911, 4.56991281471844814515420537170, 4.79715310947635289487596726359, 5.22664482048305214795586640318, 6.15140925908559062973423211653, 6.76592262620591793240938009722, 7.30506309695120754539725350015, 8.211123526792956143008172075081