| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 0.585·11-s + 12-s + 15-s + 16-s + 1.41·17-s + 18-s − 2.82·19-s + 20-s + 0.585·22-s + 4.82·23-s + 24-s + 25-s + 27-s + 8.24·29-s + 30-s + 5.07·31-s + 32-s + 0.585·33-s + 1.41·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.176·11-s + 0.288·12-s + 0.258·15-s + 0.250·16-s + 0.342·17-s + 0.235·18-s − 0.648·19-s + 0.223·20-s + 0.124·22-s + 1.00·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s + 1.53·29-s + 0.182·30-s + 0.910·31-s + 0.176·32-s + 0.101·33-s + 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.634238794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.634238794\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 4.58T + 43T^{2} \) |
| 47 | \( 1 + 9.07T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 2.48T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 7.89T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−9.569271893826653397485253885365, −8.612864607819696475975543657545, −7.964875640907786813228576752465, −6.82345703857196735616285400853, −6.36133397567610942126946200948, −5.18342862249450720149880939247, −4.49434603620725984952838538197, −3.36562541094190393038681590456, −2.58882474574882125665155700551, −1.38623849376575658283960449038,
1.38623849376575658283960449038, 2.58882474574882125665155700551, 3.36562541094190393038681590456, 4.49434603620725984952838538197, 5.18342862249450720149880939247, 6.36133397567610942126946200948, 6.82345703857196735616285400853, 7.964875640907786813228576752465, 8.612864607819696475975543657545, 9.569271893826653397485253885365
