| L(s) = 1 | − 0.329·2-s + 3-s − 1.89·4-s − 0.329·6-s + 3.70·7-s + 1.28·8-s + 9-s − 3.31·11-s − 1.89·12-s + 13-s − 1.21·14-s + 3.36·16-s − 4.36·17-s − 0.329·18-s + 5.21·19-s + 3.70·21-s + 1.09·22-s + 4.92·23-s + 1.28·24-s − 0.329·26-s + 27-s − 7.00·28-s + 7.78·29-s + 0.0981·31-s − 3.67·32-s − 3.31·33-s + 1.43·34-s + ⋯ |
| L(s) = 1 | − 0.232·2-s + 0.577·3-s − 0.945·4-s − 0.134·6-s + 1.39·7-s + 0.453·8-s + 0.333·9-s − 0.998·11-s − 0.546·12-s + 0.277·13-s − 0.325·14-s + 0.840·16-s − 1.05·17-s − 0.0776·18-s + 1.19·19-s + 0.807·21-s + 0.232·22-s + 1.02·23-s + 0.261·24-s − 0.0645·26-s + 0.192·27-s − 1.32·28-s + 1.44·29-s + 0.0176·31-s − 0.648·32-s − 0.576·33-s + 0.246·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.611533597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611533597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.329T + 2T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 - 4.92T + 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 - 0.0981T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 + 0.749T + 41T^{2} \) |
| 43 | \( 1 + 3.78T + 43T^{2} \) |
| 47 | \( 1 - 5.67T + 47T^{2} \) |
| 53 | \( 1 + 2.19T + 53T^{2} \) |
| 59 | \( 1 - 0.108T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 9.56T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + 4.10T + 97T^{2} \) |
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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
−9.908068380432167072353774614495, −9.000992232119504585911682015980, −8.331365468099961909198289874286, −7.87441947577561285035513089338, −6.86560203795572226861270782405, −5.19592726258249415993214283913, −4.91984950666240778553753211244, −3.77819648661091037160072013830, −2.48086498105810295740419761076, −1.09236781547544356470625201983,
1.09236781547544356470625201983, 2.48086498105810295740419761076, 3.77819648661091037160072013830, 4.91984950666240778553753211244, 5.19592726258249415993214283913, 6.86560203795572226861270782405, 7.87441947577561285035513089338, 8.331365468099961909198289874286, 9.000992232119504585911682015980, 9.908068380432167072353774614495
