| L(s) = 1 | + 1.52·2-s + 0.0640·3-s + 0.340·4-s − 5-s + 0.0979·6-s − 7-s − 2.53·8-s − 2.99·9-s − 1.52·10-s − 6.33·11-s + 0.0217·12-s − 2.98·13-s − 1.52·14-s − 0.0640·15-s − 4.56·16-s − 3.87·17-s − 4.58·18-s + 0.248·19-s − 0.340·20-s − 0.0640·21-s − 9.68·22-s + 4.03·23-s − 0.162·24-s + 25-s − 4.57·26-s − 0.383·27-s − 0.340·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.0369·3-s + 0.170·4-s − 0.447·5-s + 0.0399·6-s − 0.377·7-s − 0.897·8-s − 0.998·9-s − 0.483·10-s − 1.90·11-s + 0.00628·12-s − 0.829·13-s − 0.408·14-s − 0.0165·15-s − 1.14·16-s − 0.939·17-s − 1.08·18-s + 0.0569·19-s − 0.0760·20-s − 0.0139·21-s − 2.06·22-s + 0.840·23-s − 0.0331·24-s + 0.200·25-s − 0.896·26-s − 0.0739·27-s − 0.0642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5995764961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5995764961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.52T + 2T^{2} \) |
| 3 | \( 1 - 0.0640T + 3T^{2} \) |
| 11 | \( 1 + 6.33T + 11T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 - 0.248T + 19T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 - 6.29T + 31T^{2} \) |
| 37 | \( 1 + 1.45T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 0.165T + 47T^{2} \) |
| 53 | \( 1 + 3.32T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 2.79T + 61T^{2} \) |
| 67 | \( 1 + 5.18T + 67T^{2} \) |
| 71 | \( 1 + 6.62T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−7.81735164574476568916611454238, −7.02641129685718558351484405408, −6.27014652950696845797360395486, −5.51925845616187909603367952879, −4.97606208501831424106105829080, −4.50872451247910400288537648091, −3.41635396491216525656080269696, −2.84987943614847551358092469076, −2.34733341553714653419982704351, −0.29456754690664470025479849517,
0.29456754690664470025479849517, 2.34733341553714653419982704351, 2.84987943614847551358092469076, 3.41635396491216525656080269696, 4.50872451247910400288537648091, 4.97606208501831424106105829080, 5.51925845616187909603367952879, 6.27014652950696845797360395486, 7.02641129685718558351484405408, 7.81735164574476568916611454238
