| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 2·11-s + 12-s + 13-s + 16-s − 2·17-s + 18-s + 8·19-s + 2·22-s + 24-s − 5·25-s + 26-s + 27-s − 2·29-s + 10·31-s + 32-s + 2·33-s − 2·34-s + 36-s + 8·38-s + 39-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s + 0.426·22-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s + 1.29·38-s + 0.160·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.771644914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.771644914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−7.80133481495295843322819522442, −7.08745627395774677014389752993, −6.43696932264732667314272969259, −5.73167100334378367840725926975, −4.94120568059144299125201836565, −4.20734984483792387552922138060, −3.51193588232521867230999066289, −2.86600164880482890665807550991, −1.92860122178507931178977174687, −0.994033076292022934540922513142,
0.994033076292022934540922513142, 1.92860122178507931178977174687, 2.86600164880482890665807550991, 3.51193588232521867230999066289, 4.20734984483792387552922138060, 4.94120568059144299125201836565, 5.73167100334378367840725926975, 6.43696932264732667314272969259, 7.08745627395774677014389752993, 7.80133481495295843322819522442
