| L(s) = 1 | − 0.959·2-s − 7.07·4-s − 10.2·5-s + 14.4·8-s + 9.87·10-s + 28.3·11-s − 5.96·13-s + 42.7·16-s + 104.·17-s − 34.0·19-s + 72.8·20-s − 27.1·22-s − 103.·23-s − 19.1·25-s + 5.72·26-s − 195.·29-s − 9.17·31-s − 156.·32-s − 100.·34-s + 245.·37-s + 32.6·38-s − 148.·40-s − 366.·41-s − 366.·43-s − 200.·44-s + 99.4·46-s − 244.·47-s + ⋯ |
| L(s) = 1 | − 0.339·2-s − 0.884·4-s − 0.920·5-s + 0.639·8-s + 0.312·10-s + 0.776·11-s − 0.127·13-s + 0.667·16-s + 1.49·17-s − 0.411·19-s + 0.814·20-s − 0.263·22-s − 0.939·23-s − 0.152·25-s + 0.0431·26-s − 1.25·29-s − 0.0531·31-s − 0.866·32-s − 0.506·34-s + 1.09·37-s + 0.139·38-s − 0.588·40-s − 1.39·41-s − 1.29·43-s − 0.687·44-s + 0.318·46-s − 0.758·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8311784389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8311784389\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.959T + 8T^{2} \) |
| 5 | \( 1 + 10.2T + 125T^{2} \) |
| 11 | \( 1 - 28.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.96T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.17T + 2.97e4T^{2} \) |
| 37 | \( 1 - 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 366.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 244.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 281.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 181.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 24.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 336.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 196.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 683.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 619.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 176.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 761.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.27e3T + 9.12e5T^{2} \) |
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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
−9.352057427645012862769541472919, −8.228422299111423589581516694042, −7.978191715923465048146161206937, −7.01125628679825607009602591911, −5.86782068841014698626448530484, −4.94343684859852725027966272725, −3.94063277071945029910019992598, −3.48045965269665244402197762674, −1.69223889413001091209951861786, −0.49965696433510390071892428664,
0.49965696433510390071892428664, 1.69223889413001091209951861786, 3.48045965269665244402197762674, 3.94063277071945029910019992598, 4.94343684859852725027966272725, 5.86782068841014698626448530484, 7.01125628679825607009602591911, 7.978191715923465048146161206937, 8.228422299111423589581516694042, 9.352057427645012862769541472919
