| L(s) = 1 | + 4.12·2-s − 3·3-s + 9·4-s − 13.4·5-s − 12.3·6-s + 31.4·7-s + 4.12·8-s + 9·9-s − 55.4·10-s − 40.4·11-s − 27·12-s + 129.·14-s + 40.3·15-s − 55·16-s − 43.1·17-s + 37.1·18-s + 26.9·19-s − 120.·20-s − 94.2·21-s − 166.·22-s − 19.0·23-s − 12.3·24-s + 55.6·25-s − 27·27-s + 282.·28-s − 154.·29-s + 166.·30-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.577·3-s + 1.12·4-s − 1.20·5-s − 0.841·6-s + 1.69·7-s + 0.182·8-s + 0.333·9-s − 1.75·10-s − 1.11·11-s − 0.649·12-s + 2.47·14-s + 0.694·15-s − 0.859·16-s − 0.615·17-s + 0.485·18-s + 0.325·19-s − 1.35·20-s − 0.979·21-s − 1.61·22-s − 0.172·23-s − 0.105·24-s + 0.445·25-s − 0.192·27-s + 1.90·28-s − 0.986·29-s + 1.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.12T + 8T^{2} \) |
| 5 | \( 1 + 13.4T + 125T^{2} \) |
| 7 | \( 1 - 31.4T + 343T^{2} \) |
| 11 | \( 1 + 40.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 43.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 43.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 47.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 590.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 541.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 230.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 925.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.56e3T + 9.12e5T^{2} \) |
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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
−10.84532422814685111421466728304, −9.016736029592683740773621508481, −7.80682199825930562728990412766, −7.38172216807519640360987604748, −5.90142407623043661802220153470, −5.03299782136130514998042208613, −4.50493939340056767701065161529, −3.51448748757758343868187459525, −1.99114258243882634064201701255, 0,
1.99114258243882634064201701255, 3.51448748757758343868187459525, 4.50493939340056767701065161529, 5.03299782136130514998042208613, 5.90142407623043661802220153470, 7.38172216807519640360987604748, 7.80682199825930562728990412766, 9.016736029592683740773621508481, 10.84532422814685111421466728304
