| L(s) = 1 | + 3-s − 2·5-s + 7-s − 2·9-s − 7·13-s − 2·15-s + 19-s + 21-s + 23-s − 25-s − 5·27-s + 8·29-s + 8·31-s − 2·35-s − 6·37-s − 7·39-s − 6·41-s − 10·43-s + 4·45-s + 4·47-s + 49-s − 4·53-s + 57-s + 11·59-s + 10·61-s − 2·63-s + 14·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s − 1.94·13-s − 0.516·15-s + 0.229·19-s + 0.218·21-s + 0.208·23-s − 1/5·25-s − 0.962·27-s + 1.48·29-s + 1.43·31-s − 0.338·35-s − 0.986·37-s − 1.12·39-s − 0.937·41-s − 1.52·43-s + 0.596·45-s + 0.583·47-s + 1/7·49-s − 0.549·53-s + 0.132·57-s + 1.43·59-s + 1.28·61-s − 0.251·63-s + 1.73·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.79960102851999, −14.16395852741346, −13.81282610796702, −13.31050425984924, −12.39705492604870, −12.11091757286354, −11.69747829247303, −11.33591495432300, −10.43737508862721, −10.02144420921533, −9.594899554942834, −8.774816593794529, −8.325577742570663, −8.055805119922786, −7.378326373348937, −6.946494373584787, −6.326825315023668, −5.384111331236747, −4.927132219300011, −4.513410082419057, −3.646288139622087, −3.143575568355134, −2.502318118434692, −1.975140899178540, −0.8005049248881888, 0,
0.8005049248881888, 1.975140899178540, 2.502318118434692, 3.143575568355134, 3.646288139622087, 4.513410082419057, 4.927132219300011, 5.384111331236747, 6.326825315023668, 6.946494373584787, 7.378326373348937, 8.055805119922786, 8.325577742570663, 8.774816593794529, 9.594899554942834, 10.02144420921533, 10.43737508862721, 11.33591495432300, 11.69747829247303, 12.11091757286354, 12.39705492604870, 13.31050425984924, 13.81282610796702, 14.16395852741346, 14.79960102851999
