| L(s) = 1 | + (−1.18 − 1.26i)3-s − 5-s + (2 − 1.73i)7-s + (−0.186 + 2.99i)9-s + 2.52i·11-s − 4.10i·13-s + (1.18 + 1.26i)15-s − 4.37·17-s + 3.46i·19-s + (−4.55 − 0.469i)21-s − 8.51i·23-s + 25-s + (4.00 − 3.31i)27-s + 0.939i·29-s − 3.46i·31-s + ⋯ |
| L(s) = 1 | + (−0.684 − 0.728i)3-s − 0.447·5-s + (0.755 − 0.654i)7-s + (−0.0620 + 0.998i)9-s + 0.761i·11-s − 1.13i·13-s + (0.306 + 0.325i)15-s − 1.06·17-s + 0.794i·19-s + (−0.994 − 0.102i)21-s − 1.77i·23-s + 0.200·25-s + (0.769 − 0.638i)27-s + 0.174i·29-s − 0.622i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4650662150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4650662150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.18 + 1.26i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 13 | \( 1 + 4.10iT - 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 8.51iT - 23T^{2} \) |
| 29 | \( 1 - 0.939iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 6.74T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 + 1.62T + 47T^{2} \) |
| 53 | \( 1 + 1.87iT - 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 + 0.294iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−8.588951721677793076169865243255, −8.064729995550797449233201716929, −7.30891671690519249681534004517, −6.69636039483119237581559789150, −5.70235346437091606751674643579, −4.77291368670899057624532000033, −4.14811121578050420646318548785, −2.62264528870295132547606370390, −1.49825432010313750208832753898, −0.19755090751280059143772401288,
1.59079172840653522199108656460, 3.05233429595434281112138682150, 4.09337936399170725971601456687, 4.82573522482177886361665811526, 5.56723507297850865411845341210, 6.46899672759348202238310268157, 7.27264436850529632753427018499, 8.423992208445556341664672903953, 9.029090889168201574344217820571, 9.584707177893782108666534725155
