| L(s) = 1 | − 2·3-s − 3·5-s + 7-s + 9-s + 3·11-s + 4·13-s + 6·15-s − 3·17-s + 19-s − 2·21-s + 4·25-s + 4·27-s − 6·29-s + 4·31-s − 6·33-s − 3·35-s − 2·37-s − 8·39-s − 6·41-s − 43-s − 3·45-s + 3·47-s − 6·49-s + 6·51-s − 12·53-s − 9·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 1.54·15-s − 0.727·17-s + 0.229·19-s − 0.436·21-s + 4/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.04·33-s − 0.507·35-s − 0.328·37-s − 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s − 6/7·49-s + 0.840·51-s − 1.64·53-s − 1.21·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.199158339911279140575946730378, −8.462914393396165009068265832341, −7.65894969899749366452649043261, −6.68601933176938591059893570713, −6.07700564140158959836320975863, −4.96903618429312636799150610379, −4.19306407725294924458030580762, −3.33866387034377157662854676826, −1.39430917521719853589747801849, 0,
1.39430917521719853589747801849, 3.33866387034377157662854676826, 4.19306407725294924458030580762, 4.96903618429312636799150610379, 6.07700564140158959836320975863, 6.68601933176938591059893570713, 7.65894969899749366452649043261, 8.462914393396165009068265832341, 9.199158339911279140575946730378
