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
−9.190306187189758920880489233683, −8.054978541383570711787552043036, −6.88125465627670037137869614748, −6.50377139479394739440284159292, −5.65431263003525143546369430196, −4.90673304784657360260342113845, −3.90519322068123362430686348245, −2.92498802813438917948106051184, −2.47202428435657048726624240680, −1.30388760077839179780378794379,
1.20588247267754258619604068578, 2.47375158258991812598094119226, 3.75887556355572554959240766588, 4.35071476391316458220303539859, 5.48294217580497736916931205701, 5.67890267909895910224410982971, 6.42437013497291988318308494988, 7.45365199260709808475174833154, 8.430641299377378742247000241488, 8.519857517662735942655288730132