| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 1.44·7-s + 8-s + 9-s + 2·11-s + 12-s − 2.89·13-s + 1.44·14-s + 16-s − 5.44·17-s + 18-s + 6.89·19-s + 1.44·21-s + 2·22-s + 23-s + 24-s − 2.89·26-s + 27-s + 1.44·28-s + 5·29-s + 2·31-s + 32-s + 2·33-s − 5.44·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.547·7-s + 0.353·8-s + 0.333·9-s + 0.603·11-s + 0.288·12-s − 0.804·13-s + 0.387·14-s + 0.250·16-s − 1.32·17-s + 0.235·18-s + 1.58·19-s + 0.316·21-s + 0.426·22-s + 0.208·23-s + 0.204·24-s − 0.568·26-s + 0.192·27-s + 0.273·28-s + 0.928·29-s + 0.359·31-s + 0.176·32-s + 0.348·33-s − 0.934·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.175085035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.175085035\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 3.89T + 47T^{2} \) |
| 53 | \( 1 - 0.898T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 2.10T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 2.55T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.475492291105083525616654354657, −7.80097561727077035284086314981, −7.05003196700615188676304261467, −6.44584906398786098081397978031, −5.37657256796722276789367359926, −4.67214108378694713824118614469, −4.03881207677055639041209228268, −2.98381925033245147843305235949, −2.28963121165519739994752084608, −1.14191753275515505407279630527,
1.14191753275515505407279630527, 2.28963121165519739994752084608, 2.98381925033245147843305235949, 4.03881207677055639041209228268, 4.67214108378694713824118614469, 5.37657256796722276789367359926, 6.44584906398786098081397978031, 7.05003196700615188676304261467, 7.80097561727077035284086314981, 8.475492291105083525616654354657
