| L(s) = 1 | − 3-s + 9-s − 5·13-s − 17-s + 19-s − 6·23-s − 5·25-s − 27-s + 6·29-s + 5·31-s + 7·37-s + 5·39-s − 6·41-s + 43-s − 6·47-s + 51-s − 6·53-s − 57-s + 10·61-s − 5·67-s + 6·69-s − 6·71-s − 73-s + 5·75-s − 79-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.38·13-s − 0.242·17-s + 0.229·19-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s + 0.898·31-s + 1.15·37-s + 0.800·39-s − 0.937·41-s + 0.152·43-s − 0.875·47-s + 0.140·51-s − 0.824·53-s − 0.132·57-s + 1.28·61-s − 0.610·67-s + 0.722·69-s − 0.712·71-s − 0.117·73-s + 0.577·75-s − 0.112·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.44751020749246, −13.03514106647159, −12.46063235269715, −11.91668609773846, −11.78590599530209, −11.31680060446971, −10.51543463779827, −10.19593774426849, −9.694136353182722, −9.511487228684919, −8.587041290860014, −8.175040535448408, −7.643432411623835, −7.252077556284195, −6.522809287612857, −6.229553243812408, −5.672287571308382, −4.972857830048829, −4.659382816768606, −4.134917866988873, −3.433197733994057, −2.717273660278312, −2.199723220279483, −1.556205763739112, −0.6691591520724567, 0,
0.6691591520724567, 1.556205763739112, 2.199723220279483, 2.717273660278312, 3.433197733994057, 4.134917866988873, 4.659382816768606, 4.972857830048829, 5.672287571308382, 6.229553243812408, 6.522809287612857, 7.252077556284195, 7.643432411623835, 8.175040535448408, 8.587041290860014, 9.511487228684919, 9.694136353182722, 10.19593774426849, 10.51543463779827, 11.31680060446971, 11.78590599530209, 11.91668609773846, 12.46063235269715, 13.03514106647159, 13.44751020749246
