| L(s) = 1 | + 3.10·2-s + 8.39·3-s + 1.65·4-s − 15.9·5-s + 26.0·6-s − 19.7·8-s + 43.5·9-s − 49.7·10-s + 25.1·11-s + 13.8·12-s + 75.9·13-s − 134.·15-s − 74.4·16-s − 57.7·17-s + 135.·18-s − 119.·19-s − 26.4·20-s + 78.1·22-s − 45.9·23-s − 165.·24-s + 130.·25-s + 235.·26-s + 138.·27-s − 103.·29-s − 417.·30-s + 116.·31-s − 73.6·32-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 1.61·3-s + 0.206·4-s − 1.43·5-s + 1.77·6-s − 0.871·8-s + 1.61·9-s − 1.57·10-s + 0.689·11-s + 0.333·12-s + 1.62·13-s − 2.31·15-s − 1.16·16-s − 0.823·17-s + 1.77·18-s − 1.44·19-s − 0.295·20-s + 0.757·22-s − 0.417·23-s − 1.40·24-s + 1.04·25-s + 1.77·26-s + 0.988·27-s − 0.664·29-s − 2.54·30-s + 0.675·31-s − 0.407·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 - 3.10T + 8T^{2} \) |
| 3 | \( 1 - 8.39T + 27T^{2} \) |
| 5 | \( 1 + 15.9T + 125T^{2} \) |
| 11 | \( 1 - 25.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 171.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 524.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 655.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 630.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 812.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 314.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 730.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 412.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 526.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 983.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 382.T + 9.12e5T^{2} \) |
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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
−8.460228226457277248520200366305, −7.87847240926381631097163584243, −6.73490656567175375989053531532, −6.15172928215818814039138628704, −4.56605402552194305162618355739, −4.09439350483647474783245695350, −3.60399774209383348888872351736, −2.88629833173541608844427755699, −1.65850351446287686651805870166, 0,
1.65850351446287686651805870166, 2.88629833173541608844427755699, 3.60399774209383348888872351736, 4.09439350483647474783245695350, 4.56605402552194305162618355739, 6.15172928215818814039138628704, 6.73490656567175375989053531532, 7.87847240926381631097163584243, 8.460228226457277248520200366305
