| L(s) = 1 | + 3-s + 2·5-s − 2·7-s + 9-s + 13-s + 2·15-s − 2·21-s − 25-s + 27-s − 2·29-s + 2·31-s − 4·35-s + 39-s − 6·41-s − 4·43-s + 2·45-s − 6·47-s − 3·49-s − 10·53-s + 14·59-s + 10·61-s − 2·63-s + 2·65-s − 4·67-s − 4·71-s − 4·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.516·15-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.676·35-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 0.875·47-s − 3/7·49-s − 1.37·53-s + 1.82·59-s + 1.28·61-s − 0.251·63-s + 0.248·65-s − 0.488·67-s − 0.474·71-s − 0.468·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.36938158803108, −13.00322295146742, −12.75431905559724, −11.93773064516714, −11.58625215181441, −11.02301944867452, −10.27592112683304, −10.04428177381109, −9.662398637118196, −9.171114246839816, −8.691742681116384, −8.176660775300896, −7.730817239415821, −6.957355784225827, −6.648021079799167, −6.169481829581940, −5.592265327064001, −5.122892712113839, −4.447146967078978, −3.837813973035838, −3.214141155844019, −2.927294307527114, −1.993439118311811, −1.803618370791755, −0.8972296680682151, 0,
0.8972296680682151, 1.803618370791755, 1.993439118311811, 2.927294307527114, 3.214141155844019, 3.837813973035838, 4.447146967078978, 5.122892712113839, 5.592265327064001, 6.169481829581940, 6.648021079799167, 6.957355784225827, 7.730817239415821, 8.176660775300896, 8.691742681116384, 9.171114246839816, 9.662398637118196, 10.04428177381109, 10.27592112683304, 11.02301944867452, 11.58625215181441, 11.93773064516714, 12.75431905559724, 13.00322295146742, 13.36938158803108
