| L(s) = 1 | + 5-s − 11-s − i·19-s + (1 − i)23-s + 25-s + i·29-s + 31-s + (1 + i)37-s + 41-s + (1 − i)43-s + (−1 − i)47-s − i·49-s − 55-s + i·59-s + (1 + i)67-s + ⋯ |
| L(s) = 1 | + 5-s − 11-s − i·19-s + (1 − i)23-s + 25-s + i·29-s + 31-s + (1 + i)37-s + 41-s + (1 − i)43-s + (−1 − i)47-s − i·49-s − 55-s + i·59-s + (1 + i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.483483949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483483949\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + (1 + i)T + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + (1 - i)T - iT^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−8.738331015602951732028981966480, −8.271172739149102188118703992782, −7.07693253781228957577916211209, −6.70195060112868478299087943806, −5.64694940891030320170253981809, −5.11380009412380131379549430652, −4.31271459933775652156159640621, −2.85041318677267347682328820602, −2.49390221519112733229272636961, −1.07000138614464741192678716173,
1.25014256384810785344034605294, 2.38626684902158955179129635607, 3.08618063707521553158278071417, 4.32949296382578035542719194848, 5.13023926277635098893184360445, 5.93204099044015984028090742942, 6.35481563452777511878332141853, 7.65849991331762784159655355608, 7.86069572770182248889146000039, 9.073421811241552927831966085376
