| L(s) = 1 | − 4.97·2-s + 3·3-s + 16.7·4-s − 5·5-s − 14.9·6-s − 5.48·7-s − 43.3·8-s + 9·9-s + 24.8·10-s + 50.1·12-s − 24.5·13-s + 27.2·14-s − 15·15-s + 81.6·16-s + 59.3·17-s − 44.7·18-s − 5.89·19-s − 83.5·20-s − 16.4·21-s − 68.4·23-s − 129.·24-s + 25·25-s + 122.·26-s + 27·27-s − 91.6·28-s + 265.·29-s + 74.5·30-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.08·4-s − 0.447·5-s − 1.01·6-s − 0.296·7-s − 1.91·8-s + 0.333·9-s + 0.786·10-s + 1.20·12-s − 0.524·13-s + 0.520·14-s − 0.258·15-s + 1.27·16-s + 0.847·17-s − 0.585·18-s − 0.0711·19-s − 0.934·20-s − 0.170·21-s − 0.620·23-s − 1.10·24-s + 0.200·25-s + 0.921·26-s + 0.192·27-s − 0.618·28-s + 1.69·29-s + 0.453·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7812256604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7812256604\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.97T + 8T^{2} \) |
| 7 | \( 1 + 5.48T + 343T^{2} \) |
| 13 | \( 1 + 24.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.89T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 424.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 339.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 662.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 313.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 153.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 153.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 403.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 959.T + 9.12e5T^{2} \) |
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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
−8.901677878101483164732545978261, −8.146845535110716473013041916255, −7.68155212738953520809810581499, −6.93837220696184295005558493422, −6.13353153738808215739894937211, −4.78658643954611606608651611457, −3.50939886386724847681186909810, −2.64072958350920720113026199117, −1.61778701123934422766480284896, −0.53196175745247068709289621720,
0.53196175745247068709289621720, 1.61778701123934422766480284896, 2.64072958350920720113026199117, 3.50939886386724847681186909810, 4.78658643954611606608651611457, 6.13353153738808215739894937211, 6.93837220696184295005558493422, 7.68155212738953520809810581499, 8.146845535110716473013041916255, 8.901677878101483164732545978261
