| L(s) = 1 | − 2-s + 4-s − 0.102·5-s + 0.350·7-s − 8-s + 0.102·10-s − 2.02·11-s − 0.0298·13-s − 0.350·14-s + 16-s + 3.57·17-s − 2.36·19-s − 0.102·20-s + 2.02·22-s + 3.59·23-s − 4.98·25-s + 0.0298·26-s + 0.350·28-s + 4.55·29-s + 0.839·31-s − 32-s − 3.57·34-s − 0.0357·35-s − 3.75·37-s + 2.36·38-s + 0.102·40-s − 1.43·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.0456·5-s + 0.132·7-s − 0.353·8-s + 0.0322·10-s − 0.611·11-s − 0.00828·13-s − 0.0936·14-s + 0.250·16-s + 0.867·17-s − 0.542·19-s − 0.0228·20-s + 0.432·22-s + 0.748·23-s − 0.997·25-s + 0.00586·26-s + 0.0662·28-s + 0.846·29-s + 0.150·31-s − 0.176·32-s − 0.613·34-s − 0.00604·35-s − 0.617·37-s + 0.383·38-s + 0.0161·40-s − 0.224·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 0.102T + 5T^{2} \) |
| 7 | \( 1 - 0.350T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 0.0298T + 13T^{2} \) |
| 17 | \( 1 - 3.57T + 17T^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 - 0.839T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 - 9.78T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 - 7.96T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 9.62T + 83T^{2} \) |
| 89 | \( 1 - 0.871T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.62751790056413715487769426321, −6.89930205254604411929252684839, −6.24871003387657886628815197968, −5.41245346360324862346868455995, −4.80648748669115333319898359228, −3.74461261387541004853030325334, −2.98740636948925897058116724077, −2.11922667824200519163661170152, −1.18132559230621461705687492317, 0,
1.18132559230621461705687492317, 2.11922667824200519163661170152, 2.98740636948925897058116724077, 3.74461261387541004853030325334, 4.80648748669115333319898359228, 5.41245346360324862346868455995, 6.24871003387657886628815197968, 6.89930205254604411929252684839, 7.62751790056413715487769426321
