| L(s) = 1 | − 2·2-s − 3-s + 4·4-s + 2·6-s + 25·7-s − 8·8-s − 26·9-s + 9·11-s − 4·12-s + 76·13-s − 50·14-s + 16·16-s + 24·17-s + 52·18-s − 40·19-s − 25·21-s − 18·22-s + 72·23-s + 8·24-s − 152·26-s + 53·27-s + 100·28-s + 60·29-s + 26·31-s − 32·32-s − 9·33-s − 48·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s + 0.136·6-s + 1.34·7-s − 0.353·8-s − 0.962·9-s + 0.246·11-s − 0.0962·12-s + 1.62·13-s − 0.954·14-s + 1/4·16-s + 0.342·17-s + 0.680·18-s − 0.482·19-s − 0.259·21-s − 0.174·22-s + 0.652·23-s + 0.0680·24-s − 1.14·26-s + 0.377·27-s + 0.674·28-s + 0.384·29-s + 0.150·31-s − 0.176·32-s − 0.0474·33-s − 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.882500361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882500361\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 25 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 76 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 60 T + p^{3} T^{2} \) |
| 31 | \( 1 - 26 T + p^{3} T^{2} \) |
| 41 | \( 1 - 267 T + p^{3} T^{2} \) |
| 43 | \( 1 - 382 T + p^{3} T^{2} \) |
| 47 | \( 1 + 267 T + p^{3} T^{2} \) |
| 53 | \( 1 + 171 T + p^{3} T^{2} \) |
| 59 | \( 1 - 396 T + p^{3} T^{2} \) |
| 61 | \( 1 + 898 T + p^{3} T^{2} \) |
| 67 | \( 1 - 676 T + p^{3} T^{2} \) |
| 71 | \( 1 + 21 T + p^{3} T^{2} \) |
| 73 | \( 1 - 691 T + p^{3} T^{2} \) |
| 79 | \( 1 + 394 T + p^{3} T^{2} \) |
| 83 | \( 1 + 309 T + p^{3} T^{2} \) |
| 89 | \( 1 + 918 T + p^{3} T^{2} \) |
| 97 | \( 1 - 766 T + p^{3} T^{2} \) |
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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
−8.671256981009138813321586259355, −8.311879267811830507868681580692, −7.56196964813037687358498010247, −6.45059561642522673405603047024, −5.82662898471647025123710423700, −4.92312112063743030644750809260, −3.85318144107891810928733503907, −2.73808341206429012753327004414, −1.58392572038535747000921062432, −0.77744627875992826903083019674,
0.77744627875992826903083019674, 1.58392572038535747000921062432, 2.73808341206429012753327004414, 3.85318144107891810928733503907, 4.92312112063743030644750809260, 5.82662898471647025123710423700, 6.45059561642522673405603047024, 7.56196964813037687358498010247, 8.311879267811830507868681580692, 8.671256981009138813321586259355
