| L(s) = 1 | + (−2.57 + 0.595i)7-s + 3.74i·11-s − 3.36i·13-s + 0.841·17-s − 5.59i·19-s + 2.35i·23-s + 1.41i·29-s + 8.66i·31-s − 5.15·37-s + 5.74·41-s − 3.32·43-s + 6.43·47-s + (6.29 − 3.06i)49-s − 9.64i·53-s − 12.2·59-s + ⋯ |
| L(s) = 1 | + (−0.974 + 0.224i)7-s + 1.12i·11-s − 0.931i·13-s + 0.204·17-s − 1.28i·19-s + 0.490i·23-s + 0.262i·29-s + 1.55i·31-s − 0.847·37-s + 0.896·41-s − 0.507·43-s + 0.938·47-s + (0.898 − 0.438i)49-s − 1.32i·53-s − 1.59·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6829284621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6829284621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.57 - 0.595i)T \) |
| good | 11 | \( 1 - 3.74iT - 11T^{2} \) |
| 13 | \( 1 + 3.36iT - 13T^{2} \) |
| 17 | \( 1 - 0.841T + 17T^{2} \) |
| 19 | \( 1 + 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 2.35iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 5.15T + 37T^{2} \) |
| 41 | \( 1 - 5.74T + 41T^{2} \) |
| 43 | \( 1 + 3.32T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 + 9.64iT - 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 - 3.74iT - 71T^{2} \) |
| 73 | \( 1 - 0.979iT - 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 - 12.0iT - 97T^{2} \) |
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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
−7.70697946230371262323490022714, −7.00632939690870228038379350955, −6.61465123316949917816351782839, −5.56338688191428346900660339259, −5.07764611320633674154467972947, −4.15779028440768628890411989720, −3.20628254736593589466422253890, −2.66989664074773361835177721209, −1.51799585506357877149316373308, −0.19341755173784545637414774374,
0.997648462023895398599960973991, 2.19926875121425535860511873363, 3.14103174491482909206428039043, 3.83950427908061109247517435284, 4.46719117300474745053510867605, 5.76069155778302366917653099748, 6.02071106759776016036911880089, 6.76250400536779034128758569817, 7.59828833566160547036692974306, 8.190049021207562248089301144845
