| L(s) = 1 | + 2-s − 2.21·3-s + 4-s − 0.404·5-s − 2.21·6-s + 8-s + 1.89·9-s − 0.404·10-s + 6.04·11-s − 2.21·12-s − 0.404·13-s + 0.894·15-s + 16-s − 1.80·17-s + 1.89·18-s + 5.46·19-s − 0.404·20-s + 6.04·22-s − 2.94·23-s − 2.21·24-s − 4.83·25-s − 0.404·26-s + 2.44·27-s + 29-s + 0.894·30-s − 9.48·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.27·3-s + 0.5·4-s − 0.180·5-s − 0.903·6-s + 0.353·8-s + 0.631·9-s − 0.127·10-s + 1.82·11-s − 0.638·12-s − 0.112·13-s + 0.231·15-s + 0.250·16-s − 0.438·17-s + 0.446·18-s + 1.25·19-s − 0.0904·20-s + 1.28·22-s − 0.613·23-s − 0.451·24-s − 0.967·25-s − 0.0793·26-s + 0.470·27-s + 0.185·29-s + 0.163·30-s − 1.70·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.880599588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.880599588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 + 0.404T + 5T^{2} \) |
| 11 | \( 1 - 6.04T + 11T^{2} \) |
| 13 | \( 1 + 0.404T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 + 0.894T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + 0.766T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 - 1.78T + 67T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 - 1.80T + 73T^{2} \) |
| 79 | \( 1 - 3.10T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 8.65T + 97T^{2} \) |
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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
−8.915936864726148126250801755660, −7.72216083233886528685598513188, −7.00013787395527270193876620644, −6.28922623129211796664080277353, −5.77147607622564700752892372317, −4.97739346114064150731291752122, −4.11042014415655920228918629503, −3.48573324234880655266363703540, −1.99446890846902144301053859310, −0.834557481861132548908837581887,
0.834557481861132548908837581887, 1.99446890846902144301053859310, 3.48573324234880655266363703540, 4.11042014415655920228918629503, 4.97739346114064150731291752122, 5.77147607622564700752892372317, 6.28922623129211796664080277353, 7.00013787395527270193876620644, 7.72216083233886528685598513188, 8.915936864726148126250801755660
