L(s) = 1 | − 2-s − 3-s + 4-s + 4.32·5-s + 6-s + 3.19·7-s − 8-s + 9-s − 4.32·10-s + 4.37·11-s − 12-s + 0.617·13-s − 3.19·14-s − 4.32·15-s + 16-s + 17-s − 18-s + 1.19·19-s + 4.32·20-s − 3.19·21-s − 4.37·22-s + 4.91·23-s + 24-s + 13.7·25-s − 0.617·26-s − 27-s + 3.19·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.93·5-s + 0.408·6-s + 1.20·7-s − 0.353·8-s + 0.333·9-s − 1.36·10-s + 1.32·11-s − 0.288·12-s + 0.171·13-s − 0.853·14-s − 1.11·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.273·19-s + 0.967·20-s − 0.696·21-s − 0.933·22-s + 1.02·23-s + 0.204·24-s + 2.74·25-s − 0.121·26-s − 0.192·27-s + 0.603·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.579172252\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.579172252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 4.32T + 5T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 - 0.617T + 13T^{2} \) |
| 19 | \( 1 - 1.19T + 19T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 - 5.98T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 - 2.37T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 0.350T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 0.864T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 - 3.17T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.292563655713248723734518446678, −7.20192638981982567456402256904, −6.56586744280538494867884978022, −6.15409162586186224539587620180, −5.10359264920168518554914100964, −4.97176198166899851016518602749, −3.52292256263032488143216420811, −2.37080920451915986394013477281, −1.47155375206080000266496355911, −1.18835383015071006714047264235,
1.18835383015071006714047264235, 1.47155375206080000266496355911, 2.37080920451915986394013477281, 3.52292256263032488143216420811, 4.97176198166899851016518602749, 5.10359264920168518554914100964, 6.15409162586186224539587620180, 6.56586744280538494867884978022, 7.20192638981982567456402256904, 8.292563655713248723734518446678