| L(s) = 1 | + (−0.877 + 0.479i)2-s + (1.38 − 0.0991i)3-s + (0.540 − 0.841i)4-s + (1.42 + 1.72i)5-s + (−1.16 + 0.751i)6-s + (−1.42 + 3.81i)7-s + (−0.0713 + 0.997i)8-s + (−1.05 + 0.151i)9-s + (−2.07 − 0.833i)10-s + (−0.289 − 0.984i)11-s + (0.666 − 1.22i)12-s + (2.63 − 0.983i)13-s + (−0.579 − 4.03i)14-s + (2.14 + 2.25i)15-s + (−0.415 − 0.909i)16-s + (3.31 + 0.721i)17-s + ⋯ |
| L(s) = 1 | + (−0.620 + 0.338i)2-s + (0.800 − 0.0572i)3-s + (0.270 − 0.420i)4-s + (0.635 + 0.771i)5-s + (−0.477 + 0.306i)6-s + (−0.538 + 1.44i)7-s + (−0.0252 + 0.352i)8-s + (−0.351 + 0.0506i)9-s + (−0.656 − 0.263i)10-s + (−0.0871 − 0.296i)11-s + (0.192 − 0.352i)12-s + (0.731 − 0.272i)13-s + (−0.154 − 1.07i)14-s + (0.553 + 0.581i)15-s + (−0.103 − 0.227i)16-s + (0.804 + 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.993074 + 0.676594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.993074 + 0.676594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.877 - 0.479i)T \) |
| 5 | \( 1 + (-1.42 - 1.72i)T \) |
| 23 | \( 1 + (-2.11 + 4.30i)T \) |
| good | 3 | \( 1 + (-1.38 + 0.0991i)T + (2.96 - 0.426i)T^{2} \) |
| 7 | \( 1 + (1.42 - 3.81i)T + (-5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (0.289 + 0.984i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.63 + 0.983i)T + (9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (-3.31 - 0.721i)T + (15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (0.665 + 0.427i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.65 + 2.57i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.43 - 2.81i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-4.51 + 6.03i)T + (-10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-1.56 + 10.9i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.671 - 9.39i)T + (-42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (-6.88 + 6.88i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.40 - 1.26i)T + (40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (9.77 + 4.46i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.54 - 2.20i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (3.86 + 7.08i)T + (-36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (12.3 + 3.63i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.27 - 5.88i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (0.460 - 1.00i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (6.62 + 4.96i)T + (23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (-5.74 + 6.63i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.32 + 0.995i)T + (27.3 - 93.0i)T^{2} \) |
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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
−12.42241303910462743104527504956, −11.20855221704409198051796143169, −10.26360711497842573243910920434, −9.157426531168512874624926382073, −8.709201988645868485424600764416, −7.58960861350728680583416451832, −6.18194064242052290076933541880, −5.67332184688884853164133131858, −3.18144766553207759102928234744, −2.30834902915305764644381857306,
1.26354629854864160184819042637, 3.05607096406904452335515119446, 4.24135544530208478888651044504, 5.97218242923704266314175433447, 7.32571508688440554560579434953, 8.234952883514499765025643755806, 9.264081309466502069427270946157, 9.838849221374488946683304591299, 10.81205503800395869685103017398, 11.99232610266314129503420780659
