L(s) = 1 | + (−0.998 + 0.0627i)5-s + (−0.125 + 0.992i)9-s + (−0.486 − 0.627i)13-s + (−1.57 − 0.148i)17-s + (0.992 − 0.125i)25-s + (1.65 − 1.05i)29-s + (−1.58 − 1.07i)37-s + (−1.35 + 0.742i)41-s + (0.0627 − 0.998i)45-s + (0.587 − 0.809i)49-s + (−0.775 − 1.79i)53-s + (−1.74 − 0.961i)61-s + (0.525 + 0.595i)65-s + (1.01 + 0.0319i)73-s + (−0.968 − 0.248i)81-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0627i)5-s + (−0.125 + 0.992i)9-s + (−0.486 − 0.627i)13-s + (−1.57 − 0.148i)17-s + (0.992 − 0.125i)25-s + (1.65 − 1.05i)29-s + (−1.58 − 1.07i)37-s + (−1.35 + 0.742i)41-s + (0.0627 − 0.998i)45-s + (0.587 − 0.809i)49-s + (−0.775 − 1.79i)53-s + (−1.74 − 0.961i)61-s + (0.525 + 0.595i)65-s + (1.01 + 0.0319i)73-s + (−0.968 − 0.248i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3520423598\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3520423598\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.998 - 0.0627i)T \) |
good | 3 | \( 1 + (0.125 - 0.992i)T^{2} \) |
| 7 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 11 | \( 1 + (-0.637 + 0.770i)T^{2} \) |
| 13 | \( 1 + (0.486 + 0.627i)T + (-0.248 + 0.968i)T^{2} \) |
| 17 | \( 1 + (1.57 + 0.148i)T + (0.982 + 0.187i)T^{2} \) |
| 19 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 23 | \( 1 + (-0.844 + 0.535i)T^{2} \) |
| 29 | \( 1 + (-1.65 + 1.05i)T + (0.425 - 0.904i)T^{2} \) |
| 31 | \( 1 + (-0.187 + 0.982i)T^{2} \) |
| 37 | \( 1 + (1.58 + 1.07i)T + (0.368 + 0.929i)T^{2} \) |
| 41 | \( 1 + (1.35 - 0.742i)T + (0.535 - 0.844i)T^{2} \) |
| 43 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 47 | \( 1 + (-0.481 - 0.876i)T^{2} \) |
| 53 | \( 1 + (0.775 + 1.79i)T + (-0.684 + 0.728i)T^{2} \) |
| 59 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 61 | \( 1 + (1.74 + 0.961i)T + (0.535 + 0.844i)T^{2} \) |
| 67 | \( 1 + (-0.904 + 0.425i)T^{2} \) |
| 71 | \( 1 + (0.876 - 0.481i)T^{2} \) |
| 73 | \( 1 + (-1.01 - 0.0319i)T + (0.998 + 0.0627i)T^{2} \) |
| 79 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 83 | \( 1 + (-0.125 - 0.992i)T^{2} \) |
| 89 | \( 1 + (1.23 + 1.31i)T + (-0.0627 + 0.998i)T^{2} \) |
| 97 | \( 1 + (-0.0613 - 0.0137i)T + (0.904 + 0.425i)T^{2} \) |
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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
−8.353451751564676672206963979616, −7.71413198170724737767953506550, −6.98074834692967785963640315043, −6.32468902743415703086753460849, −5.03839980039416211450818637901, −4.77300378586493367144319566603, −3.76961480877980016277782134691, −2.84012970985911837625513693756, −1.98091092929615412590089834512, −0.19684317310263933227371416600,
1.37506656756952900579980608013, 2.74001902876682082282184823193, 3.50476452920446201471605306821, 4.42366824292623955314740557529, 4.85644051728215680771114966214, 6.12373257190280092567345674586, 6.87136974645224809439573181262, 7.17791630859979823841961134115, 8.396957661082891940117683959043, 8.737741721106932966419080238852