| L(s) = 1 | + 2.38·2-s − 3-s + 3.67·4-s − 5-s − 2.38·6-s − 5.05·7-s + 3.99·8-s + 9-s − 2.38·10-s − 3.47·11-s − 3.67·12-s − 1.57·13-s − 12.0·14-s + 15-s + 2.15·16-s + 6.22·17-s + 2.38·18-s − 4.56·19-s − 3.67·20-s + 5.05·21-s − 8.28·22-s + 1.93·23-s − 3.99·24-s + 25-s − 3.74·26-s − 27-s − 18.5·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.83·4-s − 0.447·5-s − 0.972·6-s − 1.90·7-s + 1.41·8-s + 0.333·9-s − 0.753·10-s − 1.04·11-s − 1.06·12-s − 0.436·13-s − 3.21·14-s + 0.258·15-s + 0.539·16-s + 1.51·17-s + 0.561·18-s − 1.04·19-s − 0.821·20-s + 1.10·21-s − 1.76·22-s + 0.402·23-s − 0.814·24-s + 0.200·25-s − 0.734·26-s − 0.192·27-s − 3.50·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.505449101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.505449101\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 + 4.56T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 5.97T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 0.887T + 37T^{2} \) |
| 41 | \( 1 - 8.87T + 41T^{2} \) |
| 43 | \( 1 - 4.84T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 0.985T + 53T^{2} \) |
| 59 | \( 1 - 2.96T + 59T^{2} \) |
| 61 | \( 1 - 8.53T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 6.06T + 73T^{2} \) |
| 79 | \( 1 - 6.84T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−7.63551245617354345726409424498, −7.05671593378849381422128422352, −6.40843160497165551561133238703, −5.81512022832598559767048552001, −5.28959892605370173315883873625, −4.39835210216922244804474014109, −3.76136897012584610885026214848, −2.95117265422490396580917128551, −2.51395839913564158461688674911, −0.62426408616543419943026084967,
0.62426408616543419943026084967, 2.51395839913564158461688674911, 2.95117265422490396580917128551, 3.76136897012584610885026214848, 4.39835210216922244804474014109, 5.28959892605370173315883873625, 5.81512022832598559767048552001, 6.40843160497165551561133238703, 7.05671593378849381422128422352, 7.63551245617354345726409424498
