| L(s) = 1 | + 128·2-s − 2.18e3·3-s + 1.63e4·4-s + 1.54e5·5-s − 2.79e5·6-s − 1.07e6·7-s + 2.09e6·8-s + 4.78e6·9-s + 1.97e7·10-s + 6.26e6·11-s − 3.58e7·12-s − 1.38e8·14-s − 3.37e8·15-s + 2.68e8·16-s + 6.12e8·18-s + 2.52e9·20-s + 2.35e9·21-s + 8.01e8·22-s − 4.58e9·24-s + 1.76e10·25-s − 1.04e10·27-s − 1.76e10·28-s + 2.88e10·29-s − 4.31e10·30-s + 5.47e10·31-s + 3.43e10·32-s − 1.37e10·33-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 1.97·5-s − 6-s − 1.30·7-s + 8-s + 9-s + 1.97·10-s + 0.321·11-s − 12-s − 1.30·14-s − 1.97·15-s + 16-s + 18-s + 1.97·20-s + 1.30·21-s + 0.321·22-s − 24-s + 2.89·25-s − 27-s − 1.30·28-s + 1.67·29-s − 1.97·30-s + 1.99·31-s + 32-s − 0.321·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(3.814876381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.814876381\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{7} T \) |
| 3 | \( 1 + p^{7} T \) |
| good | 5 | \( 1 - 154222 T + p^{14} T^{2} \) |
| 7 | \( 1 + 1078570 T + p^{14} T^{2} \) |
| 11 | \( 1 - 6264730 T + p^{14} T^{2} \) |
| 13 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 17 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 19 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 23 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 29 | \( 1 - 28883912350 T + p^{14} T^{2} \) |
| 31 | \( 1 - 54780848678 T + p^{14} T^{2} \) |
| 37 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 41 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 43 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 47 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 53 | \( 1 + 2292036335474 T + p^{14} T^{2} \) |
| 59 | \( 1 + 2785421000870 T + p^{14} T^{2} \) |
| 61 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 67 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 71 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 73 | \( 1 + 14145994951150 T + p^{14} T^{2} \) |
| 79 | \( 1 - 18764986040582 T + p^{14} T^{2} \) |
| 83 | \( 1 - 15670457974954 T + p^{14} T^{2} \) |
| 89 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 97 | \( 1 + 23646517050430 T + p^{14} T^{2} \) |
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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
−14.00738132394826514837773803344, −13.16391323394413152812449415583, −12.22900197257676856506816784803, −10.48881661644775498528418932123, −9.703983170321260301919604018439, −6.50803878098033311540790384865, −6.20938769224838822546921337577, −4.85121452662784415892943443514, −2.78962900876036096606874636302, −1.24531072619435450186724951757,
1.24531072619435450186724951757, 2.78962900876036096606874636302, 4.85121452662784415892943443514, 6.20938769224838822546921337577, 6.50803878098033311540790384865, 9.703983170321260301919604018439, 10.48881661644775498528418932123, 12.22900197257676856506816784803, 13.16391323394413152812449415583, 14.00738132394826514837773803344
