| L(s) = 1 | + 4.76·2-s + 14.7·4-s + 18.8·5-s + 23.8·7-s + 32.0·8-s + 89.7·10-s + 60.2·11-s + 113.·14-s + 34.9·16-s + 1.17·17-s − 29.9·19-s + 277.·20-s + 287.·22-s − 159.·23-s + 229.·25-s + 351.·28-s − 20.8·29-s + 67.2·31-s − 89.8·32-s + 5.60·34-s + 449.·35-s − 138.·37-s − 142.·38-s + 603.·40-s − 113.·41-s + 32.9·43-s + 886.·44-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.83·4-s + 1.68·5-s + 1.28·7-s + 1.41·8-s + 2.83·10-s + 1.65·11-s + 2.17·14-s + 0.545·16-s + 0.0167·17-s − 0.362·19-s + 3.10·20-s + 2.78·22-s − 1.44·23-s + 1.83·25-s + 2.37·28-s − 0.133·29-s + 0.389·31-s − 0.496·32-s + 0.0282·34-s + 2.17·35-s − 0.616·37-s − 0.610·38-s + 2.38·40-s − 0.431·41-s + 0.116·43-s + 3.03·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.67174549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.67174549\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.76T + 8T^{2} \) |
| 5 | \( 1 - 18.8T + 125T^{2} \) |
| 7 | \( 1 - 23.8T + 343T^{2} \) |
| 11 | \( 1 - 60.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 1.17T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 138.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 32.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 520.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 467.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 409.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 74.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 305.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 318.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 867.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 626.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 679.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 491.T + 9.12e5T^{2} \) |
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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
−9.181727316584012810241337919425, −8.269038782825263762955191189867, −7.00734326864259008386590609066, −6.23258694572707339105454446481, −5.82690637470923772501722362130, −4.86062618285091552489076223253, −4.30107044590115674584993331163, −3.17280659903891171428851918118, −1.85183158143769467192805031849, −1.64868969189344874894357439679,
1.64868969189344874894357439679, 1.85183158143769467192805031849, 3.17280659903891171428851918118, 4.30107044590115674584993331163, 4.86062618285091552489076223253, 5.82690637470923772501722362130, 6.23258694572707339105454446481, 7.00734326864259008386590609066, 8.269038782825263762955191189867, 9.181727316584012810241337919425
