| L(s) = 1 | − 2-s − 2.93·3-s + 4-s + 0.199·5-s + 2.93·6-s + 7-s − 8-s + 5.59·9-s − 0.199·10-s + 4.44·11-s − 2.93·12-s − 5.18·13-s − 14-s − 0.585·15-s + 16-s − 1.80·17-s − 5.59·18-s − 8.15·19-s + 0.199·20-s − 2.93·21-s − 4.44·22-s + 2.93·24-s − 4.96·25-s + 5.18·26-s − 7.61·27-s + 28-s − 6.19·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.69·3-s + 0.5·4-s + 0.0893·5-s + 1.19·6-s + 0.377·7-s − 0.353·8-s + 1.86·9-s − 0.0631·10-s + 1.34·11-s − 0.846·12-s − 1.43·13-s − 0.267·14-s − 0.151·15-s + 0.250·16-s − 0.436·17-s − 1.31·18-s − 1.87·19-s + 0.0446·20-s − 0.639·21-s − 0.948·22-s + 0.598·24-s − 0.992·25-s + 1.01·26-s − 1.46·27-s + 0.188·28-s − 1.15·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3021642225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3021642225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 - 0.199T + 5T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 + 5.18T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 + 8.15T + 19T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 + 1.88T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 - 4.00T + 61T^{2} \) |
| 67 | \( 1 + 7.08T + 67T^{2} \) |
| 71 | \( 1 - 0.0140T + 71T^{2} \) |
| 73 | \( 1 + 4.91T + 73T^{2} \) |
| 79 | \( 1 - 6.20T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 0.689T + 97T^{2} \) |
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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
−7.66698166125498335205042718349, −7.04470360741441599377058335131, −6.63293650353832725721558651007, −5.81950034703144943367483499889, −5.34009848269080684883934670267, −4.36968865103297591211000992205, −3.88052226328641630475231885263, −2.17514699342894987220361710208, −1.64344930930499182891497901134, −0.33403602960027873493651292670,
0.33403602960027873493651292670, 1.64344930930499182891497901134, 2.17514699342894987220361710208, 3.88052226328641630475231885263, 4.36968865103297591211000992205, 5.34009848269080684883934670267, 5.81950034703144943367483499889, 6.63293650353832725721558651007, 7.04470360741441599377058335131, 7.66698166125498335205042718349
