| L(s) = 1 | + 2-s + 3-s + 4-s − 0.375·5-s + 6-s + 4.48·7-s + 8-s + 9-s − 0.375·10-s + 1.75·11-s + 12-s − 1.25·13-s + 4.48·14-s − 0.375·15-s + 16-s − 17-s + 18-s + 0.881·19-s − 0.375·20-s + 4.48·21-s + 1.75·22-s + 3.19·23-s + 24-s − 4.85·25-s − 1.25·26-s + 27-s + 4.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.168·5-s + 0.408·6-s + 1.69·7-s + 0.353·8-s + 0.333·9-s − 0.118·10-s + 0.529·11-s + 0.288·12-s − 0.349·13-s + 1.19·14-s − 0.0970·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.202·19-s − 0.0840·20-s + 0.978·21-s + 0.374·22-s + 0.665·23-s + 0.204·24-s − 0.971·25-s − 0.246·26-s + 0.192·27-s + 0.847·28-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
| \(L(1)\) |
\(\approx\) |
\(5.157708557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.157708557\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.375T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 19 | \( 1 - 0.881T + 19T^{2} \) |
| 23 | \( 1 - 3.19T + 23T^{2} \) |
| 29 | \( 1 + 0.444T + 29T^{2} \) |
| 31 | \( 1 - 0.957T + 31T^{2} \) |
| 37 | \( 1 - 4.28T + 37T^{2} \) |
| 41 | \( 1 + 0.330T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 - 8.85T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 1.72T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.071937413354796853837538091656, −7.35488355675068328141300333305, −6.81825985338277663730491024673, −5.70828011252504085469598281172, −5.12301419742158639508059816235, −4.32164852166097484560637914300, −3.89141156568349809450626184203, −2.74992673039551970919989896926, −1.99952701444079296957066327294, −1.14711214674877648245304090249,
1.14711214674877648245304090249, 1.99952701444079296957066327294, 2.74992673039551970919989896926, 3.89141156568349809450626184203, 4.32164852166097484560637914300, 5.12301419742158639508059816235, 5.70828011252504085469598281172, 6.81825985338277663730491024673, 7.35488355675068328141300333305, 8.071937413354796853837538091656
