| L(s) = 1 | + 4·2-s + 16·4-s + 86·5-s + 64·8-s + 344·10-s − 34·11-s + 3·13-s + 256·16-s − 1.90e3·17-s + 1.48e3·19-s + 1.37e3·20-s − 136·22-s + 224·23-s + 4.27e3·25-s + 12·26-s + 6.50e3·29-s − 1.73e3·31-s + 1.02e3·32-s − 7.61e3·34-s − 7.63e3·37-s + 5.95e3·38-s + 5.50e3·40-s + 1.54e4·41-s + 1.84e4·43-s − 544·44-s + 896·46-s + 1.84e4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.53·5-s + 0.353·8-s + 1.08·10-s − 0.0847·11-s + 0.00492·13-s + 1/4·16-s − 1.59·17-s + 0.946·19-s + 0.769·20-s − 0.0599·22-s + 0.0882·23-s + 1.36·25-s + 0.00348·26-s + 1.43·29-s − 0.323·31-s + 0.176·32-s − 1.12·34-s − 0.916·37-s + 0.669·38-s + 0.543·40-s + 1.43·41-s + 1.52·43-s − 0.0423·44-s + 0.0624·46-s + 1.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.379672344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.379672344\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 86 T + p^{5} T^{2} \) |
| 11 | \( 1 + 34 T + p^{5} T^{2} \) |
| 13 | \( 1 - 3 T + p^{5} T^{2} \) |
| 17 | \( 1 + 112 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 1489 T + p^{5} T^{2} \) |
| 23 | \( 1 - 224 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6508 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1731 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7633 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15414 T + p^{5} T^{2} \) |
| 43 | \( 1 - 18491 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18462 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19956 T + p^{5} T^{2} \) |
| 59 | \( 1 + 31828 T + p^{5} T^{2} \) |
| 61 | \( 1 - 57654 T + p^{5} T^{2} \) |
| 67 | \( 1 + 60563 T + p^{5} T^{2} \) |
| 71 | \( 1 - 44834 T + p^{5} T^{2} \) |
| 73 | \( 1 + 20821 T + p^{5} T^{2} \) |
| 79 | \( 1 + 30531 T + p^{5} T^{2} \) |
| 83 | \( 1 - 110602 T + p^{5} T^{2} \) |
| 89 | \( 1 + 58992 T + p^{5} T^{2} \) |
| 97 | \( 1 - 119846 T + p^{5} T^{2} \) |
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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
−9.381505732498017334352561443603, −8.781340715330865038513138619024, −7.42732619370093528572541300681, −6.57511890994112369420462263220, −5.86307452314953997834974922781, −5.10660074364078447294055615182, −4.15683984339772916483016734544, −2.76252317727516102124637699455, −2.13088633506956037150618080989, −0.958677653623058506280910806501,
0.958677653623058506280910806501, 2.13088633506956037150618080989, 2.76252317727516102124637699455, 4.15683984339772916483016734544, 5.10660074364078447294055615182, 5.86307452314953997834974922781, 6.57511890994112369420462263220, 7.42732619370093528572541300681, 8.781340715330865038513138619024, 9.381505732498017334352561443603
