| L(s) = 1 | − 2.56·3-s + (2 + i)5-s + (−2 − 1.73i)7-s + 3.56·9-s + 0.972·11-s − 0.561i·13-s + (−5.12 − 2.56i)15-s − 4.43·17-s − 1.12i·19-s + (5.12 + 4.43i)21-s + 1.12·23-s + (3 + 4i)25-s − 1.43·27-s + 4.43i·29-s − 8.87·31-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + (0.894 + 0.447i)5-s + (−0.755 − 0.654i)7-s + 1.18·9-s + 0.293·11-s − 0.155i·13-s + (−1.32 − 0.661i)15-s − 1.07·17-s − 0.257i·19-s + (1.11 + 0.968i)21-s + 0.234·23-s + (0.600 + 0.800i)25-s − 0.276·27-s + 0.823i·29-s − 1.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4043073137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4043073137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 11 | \( 1 - 0.972T + 11T^{2} \) |
| 13 | \( 1 + 0.561iT - 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 + 1.12iT - 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.43iT - 29T^{2} \) |
| 31 | \( 1 + 8.87T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 + 8.87iT - 41T^{2} \) |
| 43 | \( 1 + 1.94iT - 43T^{2} \) |
| 47 | \( 1 - 0.972iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 14.2iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 7.68iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 8.87iT - 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−9.477929998113874240612247533881, −8.785887412700850441353920696077, −7.15270283504019761373703623875, −6.98909854615289562300680853798, −6.08680358957791202419565249200, −5.61211462705722739340113271105, −4.67174780852733008905724981571, −3.70515051804813614807905666136, −2.47817882140790543832255805362, −1.12992626964719746248555577777,
0.18945041844662859237317796577, 1.58174786695539408745057505933, 2.70373925225354240323637369364, 4.15328532675921338313392364633, 4.99842519293765940831894704301, 5.71098487981807773922709698247, 6.32675880610262076863080943169, 6.70952555184427470005147327348, 7.986389896872008493137799729046, 9.147286743192407371378972504779
