L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 91·5-s + 36·6-s − 33·7-s + 64·8-s + 81·9-s − 364·10-s − 91·11-s + 144·12-s − 610·13-s − 132·14-s − 819·15-s + 256·16-s − 1.83e3·17-s + 324·18-s − 361·19-s − 1.45e3·20-s − 297·21-s − 364·22-s − 3.43e3·23-s + 576·24-s + 5.15e3·25-s − 2.44e3·26-s + 729·27-s − 528·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.62·5-s + 0.408·6-s − 0.254·7-s + 0.353·8-s + 1/3·9-s − 1.15·10-s − 0.226·11-s + 0.288·12-s − 1.00·13-s − 0.179·14-s − 0.939·15-s + 1/4·16-s − 1.53·17-s + 0.235·18-s − 0.229·19-s − 0.813·20-s − 0.146·21-s − 0.160·22-s − 1.35·23-s + 0.204·24-s + 1.64·25-s − 0.707·26-s + 0.192·27-s − 0.127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 19 | \( 1 + p^{2} T \) |
good | 5 | \( 1 + 91 T + p^{5} T^{2} \) |
| 7 | \( 1 + 33 T + p^{5} T^{2} \) |
| 11 | \( 1 + 91 T + p^{5} T^{2} \) |
| 13 | \( 1 + 610 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1833 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3436 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3562 T + p^{5} T^{2} \) |
| 31 | \( 1 - 322 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7216 T + p^{5} T^{2} \) |
| 41 | \( 1 + 13664 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3701 T + p^{5} T^{2} \) |
| 47 | \( 1 - 9203 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29186 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27804 T + p^{5} T^{2} \) |
| 61 | \( 1 - 707 p T + p^{5} T^{2} \) |
| 67 | \( 1 + 19428 T + p^{5} T^{2} \) |
| 71 | \( 1 - 7040 T + p^{5} T^{2} \) |
| 73 | \( 1 - 37341 T + p^{5} T^{2} \) |
| 79 | \( 1 + 4972 T + p^{5} T^{2} \) |
| 83 | \( 1 + 71196 T + p^{5} T^{2} \) |
| 89 | \( 1 + 3654 T + p^{5} T^{2} \) |
| 97 | \( 1 - 62362 T + p^{5} T^{2} \) |
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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
−12.18154186201940451696645862947, −11.40665214775992343773469101076, −10.14901252213856620745307711736, −8.589247726013482235470834455842, −7.66204694811116789226974060714, −6.65969948246382348612165840012, −4.70678184402513013065539254176, −3.84156358943543872974532792638, −2.49591217082682398195681154641, 0,
2.49591217082682398195681154641, 3.84156358943543872974532792638, 4.70678184402513013065539254176, 6.65969948246382348612165840012, 7.66204694811116789226974060714, 8.589247726013482235470834455842, 10.14901252213856620745307711736, 11.40665214775992343773469101076, 12.18154186201940451696645862947