| L(s) = 1 | + 4.56·2-s + 12.8·4-s − 1.42·5-s + 3.82·7-s + 22.2·8-s − 6.50·10-s + 18.2·11-s + 17.4·14-s − 1.40·16-s + 93.1·17-s + 74.6·19-s − 18.3·20-s + 83.5·22-s − 7.35·23-s − 122.·25-s + 49.2·28-s + 211.·29-s − 183.·31-s − 184.·32-s + 425.·34-s − 5.44·35-s + 289.·37-s + 340.·38-s − 31.6·40-s + 131.·41-s + 394.·43-s + 235.·44-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.60·4-s − 0.127·5-s + 0.206·7-s + 0.982·8-s − 0.205·10-s + 0.501·11-s + 0.333·14-s − 0.0219·16-s + 1.32·17-s + 0.900·19-s − 0.204·20-s + 0.809·22-s − 0.0666·23-s − 0.983·25-s + 0.332·28-s + 1.35·29-s − 1.06·31-s − 1.01·32-s + 2.14·34-s − 0.0262·35-s + 1.28·37-s + 1.45·38-s − 0.125·40-s + 0.500·41-s + 1.39·43-s + 0.806·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.410758760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.410758760\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.56T + 8T^{2} \) |
| 5 | \( 1 + 1.42T + 125T^{2} \) |
| 7 | \( 1 - 3.82T + 343T^{2} \) |
| 11 | \( 1 - 18.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 93.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.35T + 1.21e4T^{2} \) |
| 29 | \( 1 - 211.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 289.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 131.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 201.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 16.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 475.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 252.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 295.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 892.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 170.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 472.T + 9.12e5T^{2} \) |
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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
−9.216759454199888948323191259371, −7.973482821508252061842823927051, −7.34811134403511924459684004525, −6.31957393711092048720125600696, −5.68507420651559449440057638372, −4.91907928323953374878309176427, −4.00797373923076357776797599773, −3.34101956829873139967439791744, −2.33595773564210964281104581541, −1.01515790810635399397965607146,
1.01515790810635399397965607146, 2.33595773564210964281104581541, 3.34101956829873139967439791744, 4.00797373923076357776797599773, 4.91907928323953374878309176427, 5.68507420651559449440057638372, 6.31957393711092048720125600696, 7.34811134403511924459684004525, 7.973482821508252061842823927051, 9.216759454199888948323191259371
