L(s) = 1 | + (1.30 + 1.30i)3-s + (0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + 2.41i·9-s + (−0.541 + 0.541i)13-s + (1.70 + 0.707i)15-s − 1.84·19-s + 1.84·21-s + (−1 − i)23-s + (0.707 − 0.707i)25-s + (−1.84 + 1.84i)27-s + (0.382 − 0.923i)35-s − 1.41·39-s + (0.923 + 2.23i)45-s − 1.00i·49-s + ⋯ |
L(s) = 1 | + (1.30 + 1.30i)3-s + (0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + 2.41i·9-s + (−0.541 + 0.541i)13-s + (1.70 + 0.707i)15-s − 1.84·19-s + 1.84·21-s + (−1 − i)23-s + (0.707 − 0.707i)25-s + (−1.84 + 1.84i)27-s + (0.382 − 0.923i)35-s − 1.41·39-s + (0.923 + 2.23i)45-s − 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.130159354\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.130159354\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.923 + 0.382i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 1.84T + T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 - 0.765T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.360167470381046417864243237670, −8.491634378082461904876701962228, −8.314454477797238770667561986655, −7.17910524557817010046985322830, −6.13441464666611674436008689649, −4.94813011624446114414288209781, −4.46122015604205483880223942863, −3.82891785088532818853878384556, −2.46647599469830836879211899381, −1.93149227660603585043539232570,
1.57739388883241993525740637278, 2.22732277193062947190906378954, 2.82591772903394585270448774798, 4.01694798831336901339960667600, 5.39951443164235891282806665855, 6.15032153446660348223053598019, 6.87465362474936734473817390348, 7.68256762225788007207850042747, 8.328087942465828633868684732252, 8.873729463810123636130653040940