| L(s) = 1 | + 12·7-s + 64·11-s − 58·13-s − 32·17-s − 136·19-s + 128·23-s − 144·29-s + 20·31-s + 18·37-s − 288·41-s + 200·43-s − 384·47-s − 199·49-s − 496·53-s − 128·59-s − 458·61-s + 496·67-s + 512·71-s + 602·73-s + 768·77-s + 1.10e3·79-s − 704·83-s − 960·89-s − 696·91-s − 206·97-s + 432·101-s + 68·103-s + ⋯ |
| L(s) = 1 | + 0.647·7-s + 1.75·11-s − 1.23·13-s − 0.456·17-s − 1.64·19-s + 1.16·23-s − 0.922·29-s + 0.115·31-s + 0.0799·37-s − 1.09·41-s + 0.709·43-s − 1.19·47-s − 0.580·49-s − 1.28·53-s − 0.282·59-s − 0.961·61-s + 0.904·67-s + 0.855·71-s + 0.965·73-s + 1.13·77-s + 1.57·79-s − 0.931·83-s − 1.14·89-s − 0.801·91-s − 0.215·97-s + 0.425·101-s + 0.0650·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 32 T + p^{3} T^{2} \) |
| 19 | \( 1 + 136 T + p^{3} T^{2} \) |
| 23 | \( 1 - 128 T + p^{3} T^{2} \) |
| 29 | \( 1 + 144 T + p^{3} T^{2} \) |
| 31 | \( 1 - 20 T + p^{3} T^{2} \) |
| 37 | \( 1 - 18 T + p^{3} T^{2} \) |
| 41 | \( 1 + 288 T + p^{3} T^{2} \) |
| 43 | \( 1 - 200 T + p^{3} T^{2} \) |
| 47 | \( 1 + 384 T + p^{3} T^{2} \) |
| 53 | \( 1 + 496 T + p^{3} T^{2} \) |
| 59 | \( 1 + 128 T + p^{3} T^{2} \) |
| 61 | \( 1 + 458 T + p^{3} T^{2} \) |
| 67 | \( 1 - 496 T + p^{3} T^{2} \) |
| 71 | \( 1 - 512 T + p^{3} T^{2} \) |
| 73 | \( 1 - 602 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1108 T + p^{3} T^{2} \) |
| 83 | \( 1 + 704 T + p^{3} T^{2} \) |
| 89 | \( 1 + 960 T + p^{3} T^{2} \) |
| 97 | \( 1 + 206 T + p^{3} T^{2} \) |
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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
−8.618278218293859275651324571229, −7.74662251039816058432314214805, −6.79807004225967862981246019130, −6.35553364883219959861061637988, −5.05160099310176226960627352392, −4.47086866469088418592073772571, −3.53986007721472732413237617244, −2.24983544318088224349999745481, −1.41156668815676932534814457528, 0,
1.41156668815676932534814457528, 2.24983544318088224349999745481, 3.53986007721472732413237617244, 4.47086866469088418592073772571, 5.05160099310176226960627352392, 6.35553364883219959861061637988, 6.79807004225967862981246019130, 7.74662251039816058432314214805, 8.618278218293859275651324571229
