| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 11-s + 12-s − 13-s + 16-s + 18-s − 3·19-s − 22-s − 7·23-s + 24-s − 26-s + 27-s − 8·29-s − 2·31-s + 32-s − 33-s + 36-s − 11·37-s − 3·38-s − 39-s − 11·41-s − 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.235·18-s − 0.688·19-s − 0.213·22-s − 1.45·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 1.48·29-s − 0.359·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s − 1.80·37-s − 0.486·38-s − 0.160·39-s − 1.71·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.32152115807614597129927686326, −7.03271907458506191188196403496, −6.03610627842806409208141826898, −5.42545615428333980722578510317, −4.65911051616813060532799752432, −3.80933976058569794819833624717, −3.35431246120768174032266436251, −2.22056784300383946507998461204, −1.76838682091167151224793845959, 0,
1.76838682091167151224793845959, 2.22056784300383946507998461204, 3.35431246120768174032266436251, 3.80933976058569794819833624717, 4.65911051616813060532799752432, 5.42545615428333980722578510317, 6.03610627842806409208141826898, 7.03271907458506191188196403496, 7.32152115807614597129927686326
