| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 12-s + 13-s − 14-s + 15-s + 16-s + 4·17-s − 18-s + 20-s + 21-s + 23-s − 24-s − 4·25-s − 26-s + 27-s + 28-s + 29-s − 30-s − 10·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.185·29-s − 0.182·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.83078792579637, −12.28828344338018, −11.75488884697899, −11.43953998171249, −10.80304965766032, −10.45792449708991, −9.939190296477413, −9.632630937266389, −9.036476860393829, −8.790866677794888, −8.239942244321500, −7.667638801625852, −7.487578711672913, −6.952080078168349, −6.215195004829536, −5.915053966420371, −5.350538820180765, −4.818008876429847, −4.158772193376300, −3.516620073860344, −3.181296973236605, −2.498267749003483, −1.822972308983993, −1.583829767985824, −0.8564059950049976, 0,
0.8564059950049976, 1.583829767985824, 1.822972308983993, 2.498267749003483, 3.181296973236605, 3.516620073860344, 4.158772193376300, 4.818008876429847, 5.350538820180765, 5.915053966420371, 6.215195004829536, 6.952080078168349, 7.487578711672913, 7.667638801625852, 8.239942244321500, 8.790866677794888, 9.036476860393829, 9.632630937266389, 9.939190296477413, 10.45792449708991, 10.80304965766032, 11.43953998171249, 11.75488884697899, 12.28828344338018, 12.83078792579637
