| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s − 4·7-s + 3·8-s + 9-s + 4·11-s − 12-s − 2·13-s + 4·14-s − 16-s − 2·17-s − 18-s − 19-s − 4·21-s − 4·22-s + 4·23-s + 3·24-s + 2·26-s + 27-s + 4·28-s − 2·29-s − 5·32-s + 4·33-s + 2·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.872·21-s − 0.852·22-s + 0.834·23-s + 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.883·32-s + 0.696·33-s + 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9700651033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9700651033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.456957912327740145299523356238, −8.942236798090976512890571214951, −8.204261797095146615633952733135, −7.02357868670760652523158856931, −6.70552074711622254156145782215, −5.38607911495288050360483967057, −4.17893424077245633552095753386, −3.54490000396289616314690825839, −2.29629701215321469443462239002, −0.76854595244344636007624940810,
0.76854595244344636007624940810, 2.29629701215321469443462239002, 3.54490000396289616314690825839, 4.17893424077245633552095753386, 5.38607911495288050360483967057, 6.70552074711622254156145782215, 7.02357868670760652523158856931, 8.204261797095146615633952733135, 8.942236798090976512890571214951, 9.456957912327740145299523356238
