| L(s) = 1 | + 0.707·2-s + 1.14·3-s − 1.49·4-s + 5-s + 0.813·6-s + 1.94·7-s − 2.47·8-s − 1.67·9-s + 0.707·10-s + 4.58·11-s − 1.72·12-s − 3.56·13-s + 1.37·14-s + 1.14·15-s + 1.24·16-s + 6.32·17-s − 1.18·18-s − 2.19·19-s − 1.49·20-s + 2.23·21-s + 3.24·22-s + 9.36·23-s − 2.84·24-s + 25-s − 2.52·26-s − 5.37·27-s − 2.91·28-s + ⋯ |
| L(s) = 1 | + 0.500·2-s + 0.663·3-s − 0.749·4-s + 0.447·5-s + 0.331·6-s + 0.733·7-s − 0.875·8-s − 0.559·9-s + 0.223·10-s + 1.38·11-s − 0.497·12-s − 0.989·13-s + 0.366·14-s + 0.296·15-s + 0.311·16-s + 1.53·17-s − 0.279·18-s − 0.504·19-s − 0.335·20-s + 0.486·21-s + 0.690·22-s + 1.95·23-s − 0.580·24-s + 0.200·25-s − 0.494·26-s − 1.03·27-s − 0.550·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.558274989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558274989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.707T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 - 9.36T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 - 3.88T + 31T^{2} \) |
| 37 | \( 1 - 0.0793T + 37T^{2} \) |
| 41 | \( 1 - 0.604T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 3.42T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 0.934T + 59T^{2} \) |
| 61 | \( 1 - 5.44T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 + 6.57T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 - 7.86T + 89T^{2} \) |
| 97 | \( 1 + 5.44T + 97T^{2} \) |
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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
−9.603310033932063802164236161216, −8.833192206381344267887512056534, −8.400093798361776945034628133704, −7.32342521159858276207153869005, −6.26672932792696632301248309324, −5.27753824067178508164871572894, −4.66373197863068178461112110536, −3.53564464466319931530208920419, −2.71994761922588908314186710169, −1.19841670604256266444005536676,
1.19841670604256266444005536676, 2.71994761922588908314186710169, 3.53564464466319931530208920419, 4.66373197863068178461112110536, 5.27753824067178508164871572894, 6.26672932792696632301248309324, 7.32342521159858276207153869005, 8.400093798361776945034628133704, 8.833192206381344267887512056534, 9.603310033932063802164236161216
