| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 3·9-s − 13-s + 14-s − 16-s + 6·17-s − 3·18-s + 4·23-s − 26-s − 28-s − 2·29-s − 4·31-s + 5·32-s + 6·34-s + 3·36-s + 10·37-s + 2·41-s + 8·43-s + 4·46-s + 49-s + 52-s + 2·53-s − 3·56-s − 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s − 0.277·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.834·23-s − 0.196·26-s − 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s + 1.64·37-s + 0.312·41-s + 1.21·43-s + 0.589·46-s + 1/7·49-s + 0.138·52-s + 0.274·53-s − 0.400·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.920118616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.920118616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.164067461048079087868657186778, −8.158775253871522966729620501963, −7.64094675378022286118513099355, −6.43290082826129774900885337637, −5.55851840137587019955463935483, −5.21444600671299633906090220383, −4.17639011719279793332592730095, −3.33685054509951812665221984253, −2.50405845676474902927875411466, −0.812572614917737503035785374863,
0.812572614917737503035785374863, 2.50405845676474902927875411466, 3.33685054509951812665221984253, 4.17639011719279793332592730095, 5.21444600671299633906090220383, 5.55851840137587019955463935483, 6.43290082826129774900885337637, 7.64094675378022286118513099355, 8.158775253871522966729620501963, 9.164067461048079087868657186778
