L(s) = 1 | + 1.45·2-s − 3·3-s − 5.88·4-s + 5·5-s − 4.35·6-s − 20.0·7-s − 20.1·8-s + 9·9-s + 7.26·10-s + 17.6·12-s − 13.0·13-s − 29.1·14-s − 15·15-s + 17.7·16-s + 38.6·17-s + 13.0·18-s + 23.6·19-s − 29.4·20-s + 60.1·21-s + 63.5·23-s + 60.5·24-s + 25·25-s − 18.9·26-s − 27·27-s + 117.·28-s + 54.4·29-s − 21.7·30-s + ⋯ |
L(s) = 1 | + 0.513·2-s − 0.577·3-s − 0.736·4-s + 0.447·5-s − 0.296·6-s − 1.08·7-s − 0.891·8-s + 0.333·9-s + 0.229·10-s + 0.425·12-s − 0.278·13-s − 0.555·14-s − 0.258·15-s + 0.278·16-s + 0.551·17-s + 0.171·18-s + 0.285·19-s − 0.329·20-s + 0.624·21-s + 0.575·23-s + 0.514·24-s + 0.200·25-s − 0.143·26-s − 0.192·27-s + 0.796·28-s + 0.348·29-s − 0.132·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.45T + 8T^{2} \) |
| 7 | \( 1 + 20.0T + 343T^{2} \) |
| 13 | \( 1 + 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 63.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 1.71T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 88.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 414.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 126.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 161.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 259.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 427.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 68.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 970.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 601.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 601.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 197.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 977.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−8.747822886148616718116338388436, −7.56751309939151830155258173984, −6.68810954253631947132212246834, −5.90797456081360760548891344755, −5.35042949705429950482030785883, −4.46183722641245440739538334002, −3.52704461853168230977992780305, −2.68738297992382123884367311562, −1.05491151753627855919594335701, 0,
1.05491151753627855919594335701, 2.68738297992382123884367311562, 3.52704461853168230977992780305, 4.46183722641245440739538334002, 5.35042949705429950482030785883, 5.90797456081360760548891344755, 6.68810954253631947132212246834, 7.56751309939151830155258173984, 8.747822886148616718116338388436