L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 216·6-s − 416·7-s + 512·8-s + 729·9-s − 3.94e3·11-s + 1.72e3·12-s + 7.97e3·13-s − 3.32e3·14-s + 4.09e3·16-s + 3.40e4·17-s + 5.83e3·18-s + 3.02e3·19-s − 1.12e4·21-s − 3.15e4·22-s + 6.65e4·23-s + 1.38e4·24-s + 6.38e4·26-s + 1.96e4·27-s − 2.66e4·28-s − 1.85e4·29-s + 2.08e5·31-s + 3.27e4·32-s − 1.06e5·33-s + 2.72e5·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.458·7-s + 0.353·8-s + 1/3·9-s − 0.894·11-s + 0.288·12-s + 1.00·13-s − 0.324·14-s + 1/4·16-s + 1.67·17-s + 0.235·18-s + 0.101·19-s − 0.264·21-s − 0.632·22-s + 1.14·23-s + 0.204·24-s + 0.712·26-s + 0.192·27-s − 0.229·28-s − 0.141·29-s + 1.25·31-s + 0.176·32-s − 0.516·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(4.219220436\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.219220436\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 416 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3948 T + p^{7} T^{2} \) |
| 13 | \( 1 - 7978 T + p^{7} T^{2} \) |
| 17 | \( 1 - 34014 T + p^{7} T^{2} \) |
| 19 | \( 1 - 3020 T + p^{7} T^{2} \) |
| 23 | \( 1 - 66528 T + p^{7} T^{2} \) |
| 29 | \( 1 + 18570 T + p^{7} T^{2} \) |
| 31 | \( 1 - 208832 T + p^{7} T^{2} \) |
| 37 | \( 1 - 301474 T + p^{7} T^{2} \) |
| 41 | \( 1 + 460998 T + p^{7} T^{2} \) |
| 43 | \( 1 + 343052 T + p^{7} T^{2} \) |
| 47 | \( 1 - 1356264 T + p^{7} T^{2} \) |
| 53 | \( 1 + 617982 T + p^{7} T^{2} \) |
| 59 | \( 1 - 939540 T + p^{7} T^{2} \) |
| 61 | \( 1 + 204298 T + p^{7} T^{2} \) |
| 67 | \( 1 - 758524 T + p^{7} T^{2} \) |
| 71 | \( 1 - 912072 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3043322 T + p^{7} T^{2} \) |
| 79 | \( 1 - 6010880 T + p^{7} T^{2} \) |
| 83 | \( 1 + 9723012 T + p^{7} T^{2} \) |
| 89 | \( 1 - 7160010 T + p^{7} T^{2} \) |
| 97 | \( 1 - 16785214 T + p^{7} T^{2} \) |
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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
−11.89231774641965247092302659031, −10.67502904929569000406994112047, −9.766553813274768128535635372820, −8.422910902463274332880794011950, −7.45603728359916676626590018140, −6.18988387808141624184858209239, −5.05720342330715308380526885355, −3.59753178652194269584160078655, −2.75957660217848854532458857270, −1.09902644728348223559498648928,
1.09902644728348223559498648928, 2.75957660217848854532458857270, 3.59753178652194269584160078655, 5.05720342330715308380526885355, 6.18988387808141624184858209239, 7.45603728359916676626590018140, 8.422910902463274332880794011950, 9.766553813274768128535635372820, 10.67502904929569000406994112047, 11.89231774641965247092302659031