| L(s) = 1 | − 0.421·2-s − 9.69·3-s − 7.82·4-s − 5·5-s + 4.08·6-s − 27.4·7-s + 6.66·8-s + 67.0·9-s + 2.10·10-s + 59.3·11-s + 75.8·12-s + 15.4·13-s + 11.5·14-s + 48.4·15-s + 59.7·16-s − 28.2·18-s + 44.9·19-s + 39.1·20-s + 266.·21-s − 24.9·22-s − 151.·23-s − 64.6·24-s + 25·25-s − 6.51·26-s − 388.·27-s + 214.·28-s + 162.·29-s + ⋯ |
| L(s) = 1 | − 0.148·2-s − 1.86·3-s − 0.977·4-s − 0.447·5-s + 0.277·6-s − 1.48·7-s + 0.294·8-s + 2.48·9-s + 0.0666·10-s + 1.62·11-s + 1.82·12-s + 0.330·13-s + 0.220·14-s + 0.834·15-s + 0.933·16-s − 0.369·18-s + 0.542·19-s + 0.437·20-s + 2.76·21-s − 0.242·22-s − 1.37·23-s − 0.549·24-s + 0.200·25-s − 0.0491·26-s − 2.76·27-s + 1.44·28-s + 1.04·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4241271041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4241271041\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.421T + 8T^{2} \) |
| 3 | \( 1 + 9.69T + 27T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 11 | \( 1 - 59.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 44.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 326.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 19.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 334.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 6.65T + 2.26e5T^{2} \) |
| 67 | \( 1 - 503.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 157.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 658.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 590.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 707.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 739.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 355.T + 9.12e5T^{2} \) |
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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
−9.414344030064161893230856244224, −8.495897169553057489610981295888, −7.23129110032072067523231986918, −6.46052959323885574510922845311, −6.04646489286400409849127545597, −5.01652939154373854046090368056, −4.11777396201318049154951781234, −3.56114599859228502509499416037, −1.24662039667085840375068708432, −0.43052693711502763806522014396,
0.43052693711502763806522014396, 1.24662039667085840375068708432, 3.56114599859228502509499416037, 4.11777396201318049154951781234, 5.01652939154373854046090368056, 6.04646489286400409849127545597, 6.46052959323885574510922845311, 7.23129110032072067523231986918, 8.495897169553057489610981295888, 9.414344030064161893230856244224
