L(s) = 1 | + (0.540 + 0.841i)2-s + (−0.415 + 0.909i)4-s + (0.905 − 1.21i)5-s + (−0.989 + 0.142i)8-s + (1.50 + 0.107i)10-s + (0.610 − 0.655i)13-s + (−0.654 − 0.755i)16-s + (0.373 + 1.71i)17-s + (0.724 + 1.32i)20-s + (−0.361 − 1.23i)25-s + (0.881 + 0.158i)26-s + (0.183 − 0.109i)29-s + (0.281 − 0.959i)32-s + (−1.24 + 1.24i)34-s + (1.81 − 0.753i)37-s + ⋯ |
L(s) = 1 | + (0.540 + 0.841i)2-s + (−0.415 + 0.909i)4-s + (0.905 − 1.21i)5-s + (−0.989 + 0.142i)8-s + (1.50 + 0.107i)10-s + (0.610 − 0.655i)13-s + (−0.654 − 0.755i)16-s + (0.373 + 1.71i)17-s + (0.724 + 1.32i)20-s + (−0.361 − 1.23i)25-s + (0.881 + 0.158i)26-s + (0.183 − 0.109i)29-s + (0.281 − 0.959i)32-s + (−1.24 + 1.24i)34-s + (1.81 − 0.753i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.897732118\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.897732118\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 3 | \( 1 \) |
| 89 | \( 1 + (0.800 - 0.599i)T \) |
good | 5 | \( 1 + (-0.905 + 1.21i)T + (-0.281 - 0.959i)T^{2} \) |
| 7 | \( 1 + (-0.877 - 0.479i)T^{2} \) |
| 11 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.610 + 0.655i)T + (-0.0713 - 0.997i)T^{2} \) |
| 17 | \( 1 + (-0.373 - 1.71i)T + (-0.909 + 0.415i)T^{2} \) |
| 19 | \( 1 + (0.800 + 0.599i)T^{2} \) |
| 23 | \( 1 + (-0.599 + 0.800i)T^{2} \) |
| 29 | \( 1 + (-0.183 + 0.109i)T + (0.479 - 0.877i)T^{2} \) |
| 31 | \( 1 + (0.599 + 0.800i)T^{2} \) |
| 37 | \( 1 + (-1.81 + 0.753i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (0.928 + 0.997i)T + (-0.0713 + 0.997i)T^{2} \) |
| 43 | \( 1 + (0.479 + 0.877i)T^{2} \) |
| 47 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 53 | \( 1 + (-0.148 + 0.398i)T + (-0.755 - 0.654i)T^{2} \) |
| 59 | \( 1 + (-0.997 - 0.0713i)T^{2} \) |
| 61 | \( 1 + (-1.50 - 1.21i)T + (0.212 + 0.977i)T^{2} \) |
| 67 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.281 + 0.959i)T^{2} \) |
| 73 | \( 1 + (0.627 - 0.544i)T + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (0.989 + 0.142i)T^{2} \) |
| 83 | \( 1 + (-0.349 + 0.936i)T^{2} \) |
| 97 | \( 1 + (0.266 + 1.85i)T + (-0.959 + 0.281i)T^{2} \) |
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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
−8.519857517662735942655288730132, −8.430641299377378742247000241488, −7.45365199260709808475174833154, −6.42437013497291988318308494988, −5.67890267909895910224410982971, −5.48294217580497736916931205701, −4.35071476391316458220303539859, −3.75887556355572554959240766588, −2.47375158258991812598094119226, −1.20588247267754258619604068578,
1.30388760077839179780378794379, 2.47202428435657048726624240680, 2.92498802813438917948106051184, 3.90519322068123362430686348245, 4.90673304784657360260342113845, 5.65431263003525143546369430196, 6.50377139479394739440284159292, 6.88125465627670037137869614748, 8.054978541383570711787552043036, 9.190306187189758920880489233683