| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 34·7-s − 8·8-s − 10·10-s − 28·11-s − 6·13-s + 68·14-s + 16·16-s − 8·17-s + 19·19-s + 20·20-s + 56·22-s + 204·23-s + 25·25-s + 12·26-s − 136·28-s − 262·29-s + 298·31-s − 32·32-s + 16·34-s − 170·35-s + 346·37-s − 38·38-s − 40·40-s + 296·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.83·7-s − 0.353·8-s − 0.316·10-s − 0.767·11-s − 0.128·13-s + 1.29·14-s + 1/4·16-s − 0.114·17-s + 0.229·19-s + 0.223·20-s + 0.542·22-s + 1.84·23-s + 1/5·25-s + 0.0905·26-s − 0.917·28-s − 1.67·29-s + 1.72·31-s − 0.176·32-s + 0.0807·34-s − 0.821·35-s + 1.53·37-s − 0.162·38-s − 0.158·40-s + 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 8 T + p^{3} T^{2} \) |
| 23 | \( 1 - 204 T + p^{3} T^{2} \) |
| 29 | \( 1 + 262 T + p^{3} T^{2} \) |
| 31 | \( 1 - 298 T + p^{3} T^{2} \) |
| 37 | \( 1 - 346 T + p^{3} T^{2} \) |
| 41 | \( 1 - 296 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 + 462 T + p^{3} T^{2} \) |
| 59 | \( 1 + 194 T + p^{3} T^{2} \) |
| 61 | \( 1 + 46 T + p^{3} T^{2} \) |
| 67 | \( 1 + 20 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 + 922 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1382 T + p^{3} T^{2} \) |
| 83 | \( 1 + 10 T + p^{3} T^{2} \) |
| 89 | \( 1 + 180 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
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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
−8.903021872086404507888833241163, −7.65656585224673359322625499825, −7.10061999752612365860068593094, −6.18175293921556114645453769552, −5.67237303554513277984629105106, −4.35330429703287774869297403215, −2.97966645679990049822244236886, −2.66879027372228306173071091929, −1.04632001395623362360105328540, 0,
1.04632001395623362360105328540, 2.66879027372228306173071091929, 2.97966645679990049822244236886, 4.35330429703287774869297403215, 5.67237303554513277984629105106, 6.18175293921556114645453769552, 7.10061999752612365860068593094, 7.65656585224673359322625499825, 8.903021872086404507888833241163
