L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s − 4·7-s − 64·8-s + 81·9-s − 100·10-s + 96·11-s − 144·12-s + 169·13-s + 16·14-s − 225·15-s + 256·16-s − 1.18e3·17-s − 324·18-s − 1.16e3·19-s + 400·20-s + 36·21-s − 384·22-s + 1.80e3·23-s + 576·24-s + 625·25-s − 676·26-s − 729·27-s − 64·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.0308·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.239·11-s − 0.288·12-s + 0.277·13-s + 0.0218·14-s − 0.258·15-s + 1/4·16-s − 0.991·17-s − 0.235·18-s − 0.742·19-s + 0.223·20-s + 0.0178·21-s − 0.169·22-s + 0.709·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.0154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| 13 | \( 1 - p^{2} T \) |
good | 7 | \( 1 + 4 T + p^{5} T^{2} \) |
| 11 | \( 1 - 96 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1182 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1168 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1800 T + p^{5} T^{2} \) |
| 29 | \( 1 - 366 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4564 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10262 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14310 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12956 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3252 T + p^{5} T^{2} \) |
| 53 | \( 1 - 5670 T + p^{5} T^{2} \) |
| 59 | \( 1 - 42384 T + p^{5} T^{2} \) |
| 61 | \( 1 - 29678 T + p^{5} T^{2} \) |
| 67 | \( 1 + 22384 T + p^{5} T^{2} \) |
| 71 | \( 1 - 3732 T + p^{5} T^{2} \) |
| 73 | \( 1 - 37658 T + p^{5} T^{2} \) |
| 79 | \( 1 + 58792 T + p^{5} T^{2} \) |
| 83 | \( 1 + 66912 T + p^{5} T^{2} \) |
| 89 | \( 1 + 132486 T + p^{5} T^{2} \) |
| 97 | \( 1 + 12910 T + p^{5} T^{2} \) |
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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
−10.04781188757488831525319068584, −9.164230501829264507189288495273, −8.358685148898557391592231938662, −7.06846325341058565086931652161, −6.39428781579830704753372038774, −5.36321388453189646763624627036, −4.08300457502574447428356925315, −2.48826794362063188637643741970, −1.29466141915594899586383310465, 0,
1.29466141915594899586383310465, 2.48826794362063188637643741970, 4.08300457502574447428356925315, 5.36321388453189646763624627036, 6.39428781579830704753372038774, 7.06846325341058565086931652161, 8.358685148898557391592231938662, 9.164230501829264507189288495273, 10.04781188757488831525319068584