| L(s) = 1 | + (1 − i)3-s − i·4-s + i·5-s − i·9-s + (−1 − i)12-s + (1 + i)15-s − 16-s + 20-s + (−1 + i)23-s − 25-s − 36-s + (−1 − i)37-s + 45-s + (1 + i)47-s + (−1 + i)48-s − i·49-s + ⋯ |
| L(s) = 1 | + (1 − i)3-s − i·4-s + i·5-s − i·9-s + (−1 − i)12-s + (1 + i)15-s − 16-s + 20-s + (−1 + i)23-s − 25-s − 36-s + (−1 − i)37-s + 45-s + (1 + i)47-s + (−1 + i)48-s − i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.201455205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201455205\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - iT \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + iT^{2} \) |
| 3 | \( 1 + (-1 + i)T - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-1 - i)T + iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 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
−10.66432270105452716538198204293, −9.822625217518263851664909433323, −9.001792083837410620921889232778, −7.912390453660281690019820361981, −7.19012575134509703633025632719, −6.41095617984693340606172319639, −5.47233173151032294161756887476, −3.82988448522228900863698851986, −2.59486025114531446469000277362, −1.68645142704929817668356953408,
2.31090309747750861747671519750, 3.53099318022625096031249839573, 4.23444460872874093496964994527, 5.10146687064404698498950396081, 6.64236480728527275663999549975, 8.066546338408269348599003525972, 8.319637643128064961365964581172, 9.193566714952607992352400124883, 9.824294338756279619198102083953, 10.88866743325709708029794175248
