| L(s) = 1 | + 2.41·2-s + 0.757·3-s − 2.17·4-s − 10.6·5-s + 1.82·6-s − 22.1·7-s − 24.5·8-s − 26.4·9-s − 25.7·10-s − 39.3·11-s − 1.64·12-s + 23.7·13-s − 53.4·14-s − 8.07·15-s − 41.9·16-s − 4.54·17-s − 63.7·18-s + 155.·19-s + 23.1·20-s − 16.7·21-s − 94.9·22-s − 41.8·23-s − 18.5·24-s − 11.4·25-s + 57.3·26-s − 40.4·27-s + 48.0·28-s + ⋯ |
| L(s) = 1 | + 0.853·2-s + 0.145·3-s − 0.271·4-s − 0.953·5-s + 0.124·6-s − 1.19·7-s − 1.08·8-s − 0.978·9-s − 0.813·10-s − 1.07·11-s − 0.0395·12-s + 0.507·13-s − 1.02·14-s − 0.138·15-s − 0.654·16-s − 0.0648·17-s − 0.835·18-s + 1.87·19-s + 0.258·20-s − 0.174·21-s − 0.920·22-s − 0.379·23-s − 0.158·24-s − 0.0914·25-s + 0.432·26-s − 0.288·27-s + 0.324·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8498506994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8498506994\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 8T^{2} \) |
| 3 | \( 1 - 0.757T + 27T^{2} \) |
| 5 | \( 1 + 10.6T + 125T^{2} \) |
| 7 | \( 1 + 22.1T + 343T^{2} \) |
| 11 | \( 1 + 39.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 57.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 175.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 402.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 227.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 673.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 800.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 222.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 524.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 515.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 358.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 829.T + 9.12e5T^{2} \) |
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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
−9.715283997546882757137838784975, −8.962349531524075762065756452323, −8.090498373686484116568022076086, −7.24866367481433427276734263291, −5.96829970745027754995465441768, −5.46452305861643425840641897542, −4.27042904819367828935823896248, −3.29790236869070358502582208775, −2.89048670089084913087741183830, −0.42063794587729269403899150732,
0.42063794587729269403899150732, 2.89048670089084913087741183830, 3.29790236869070358502582208775, 4.27042904819367828935823896248, 5.46452305861643425840641897542, 5.96829970745027754995465441768, 7.24866367481433427276734263291, 8.090498373686484116568022076086, 8.962349531524075762065756452323, 9.715283997546882757137838784975
