| L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 2·11-s − 13-s − 3·17-s + 19-s + 3·21-s + 23-s + 5·27-s + 5·29-s − 8·31-s + 2·33-s − 2·37-s + 39-s − 8·41-s + 4·43-s − 8·47-s + 2·49-s + 3·51-s − 53-s − 57-s − 15·59-s − 2·61-s + 6·63-s + 3·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s − 0.727·17-s + 0.229·19-s + 0.654·21-s + 0.208·23-s + 0.962·27-s + 0.928·29-s − 1.43·31-s + 0.348·33-s − 0.328·37-s + 0.160·39-s − 1.24·41-s + 0.609·43-s − 1.16·47-s + 2/7·49-s + 0.420·51-s − 0.137·53-s − 0.132·57-s − 1.95·59-s − 0.256·61-s + 0.755·63-s + 0.366·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.75024232283703, −15.28285808061148, −14.51152571878212, −14.10517645516261, −13.38269463607385, −12.99380390726918, −12.48548826120316, −11.94931295644976, −11.38600775787380, −10.81675512425511, −10.37031166991611, −9.782209675697687, −9.165204374271231, −8.697809655798648, −8.025688920345713, −7.298017582462762, −6.726243581688413, −6.248080213029625, −5.667868407288241, −5.053753550976878, −4.514151475888675, −3.507243827903830, −3.038431752464666, −2.396462378236990, −1.374856210206865, 0, 0,
1.374856210206865, 2.396462378236990, 3.038431752464666, 3.507243827903830, 4.514151475888675, 5.053753550976878, 5.667868407288241, 6.248080213029625, 6.726243581688413, 7.298017582462762, 8.025688920345713, 8.697809655798648, 9.165204374271231, 9.782209675697687, 10.37031166991611, 10.81675512425511, 11.38600775787380, 11.94931295644976, 12.48548826120316, 12.99380390726918, 13.38269463607385, 14.10517645516261, 14.51152571878212, 15.28285808061148, 15.75024232283703
