L(s) = 1 | + 2-s + 1.63·3-s + 4-s + 1.63·6-s + 8-s − 0.316·9-s − 1.31·11-s + 1.63·12-s + 6.10·13-s + 16-s + 2.60·17-s − 0.316·18-s − 3.05·19-s − 1.31·22-s + 4.63·23-s + 1.63·24-s + 6.10·26-s − 5.43·27-s + 10.6·29-s − 5.65·31-s + 32-s − 2.15·33-s + 2.60·34-s − 0.316·36-s + 8.63·37-s − 3.05·38-s + 10·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.945·3-s + 0.5·4-s + 0.668·6-s + 0.353·8-s − 0.105·9-s − 0.396·11-s + 0.472·12-s + 1.69·13-s + 0.250·16-s + 0.631·17-s − 0.0746·18-s − 0.700·19-s − 0.280·22-s + 0.966·23-s + 0.334·24-s + 1.19·26-s − 1.04·27-s + 1.97·29-s − 1.01·31-s + 0.176·32-s − 0.375·33-s + 0.446·34-s − 0.0527·36-s + 1.41·37-s − 0.495·38-s + 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.152992348\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.152992348\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.63T + 3T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 3.05T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 8.63T + 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 3.27T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 + 3.27T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 2.60T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 9.15T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−8.756866709786182495476872266451, −8.226903129973287715292041881420, −7.52312592021583712172640945604, −6.44607059304817718292137913095, −5.89273455402983024099763734749, −4.89143535073419907777703123334, −3.93720044389625648617723619856, −3.20402372946152732104744306201, −2.50204672166952138292254899027, −1.23397469299771019266967023079,
1.23397469299771019266967023079, 2.50204672166952138292254899027, 3.20402372946152732104744306201, 3.93720044389625648617723619856, 4.89143535073419907777703123334, 5.89273455402983024099763734749, 6.44607059304817718292137913095, 7.52312592021583712172640945604, 8.226903129973287715292041881420, 8.756866709786182495476872266451