L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s + 2·13-s + 16-s + 2·17-s + 18-s + 19-s − 4·23-s + 24-s + 2·26-s + 27-s − 6·29-s + 4·31-s + 32-s + 2·34-s + 36-s + 6·37-s + 38-s + 2·39-s − 10·41-s − 4·43-s − 4·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.834·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.986·37-s + 0.162·38-s + 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.73133533111687, −12.47093869721125, −11.96960898947314, −11.40219898641300, −11.19729640935963, −10.39135627612567, −10.22061270876350, −9.543569085975613, −9.248557827345609, −8.561988968762288, −8.139380496570087, −7.726961343882908, −7.308804390867045, −6.659667706002761, −6.268463442954182, −5.726526249899186, −5.342057280760899, −4.557965348194942, −4.344834076229430, −3.564017256299382, −3.334577739428060, −2.762562147519610, −2.055957861892499, −1.620542359235906, −0.9673508176687162, 0,
0.9673508176687162, 1.620542359235906, 2.055957861892499, 2.762562147519610, 3.334577739428060, 3.564017256299382, 4.344834076229430, 4.557965348194942, 5.342057280760899, 5.726526249899186, 6.268463442954182, 6.659667706002761, 7.308804390867045, 7.726961343882908, 8.139380496570087, 8.561988968762288, 9.248557827345609, 9.543569085975613, 10.22061270876350, 10.39135627612567, 11.19729640935963, 11.40219898641300, 11.96960898947314, 12.47093869721125, 12.73133533111687