L(s) = 1 | − 2-s + 3-s + 4-s − 2.61·5-s − 6-s − 8-s + 9-s + 2.61·10-s − 3.55·11-s + 12-s − 13-s − 2.61·15-s + 16-s + 7.95·17-s − 18-s + 2.17·19-s − 2.61·20-s + 3.55·22-s − 2.55·23-s − 24-s + 1.82·25-s + 26-s + 27-s − 3·29-s + 2.61·30-s + 0.828·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.16·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.826·10-s − 1.07·11-s + 0.288·12-s − 0.277·13-s − 0.674·15-s + 0.250·16-s + 1.92·17-s − 0.235·18-s + 0.498·19-s − 0.584·20-s + 0.758·22-s − 0.533·23-s − 0.204·24-s + 0.365·25-s + 0.196·26-s + 0.192·27-s − 0.557·29-s + 0.477·30-s + 0.148·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 0.773T + 43T^{2} \) |
| 47 | \( 1 - 2.94T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 + 6.72T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 7.82T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 4.21T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.130858904056281548866533661730, −7.51328561520881343456728042037, −7.21474648018584727348480981586, −5.89331222566998407995159637505, −5.16102648122241101840843222641, −4.05773382380719118038019486489, −3.29947709362510016722678506907, −2.58410224238832967050439894044, −1.28003380186632762188124975126, 0,
1.28003380186632762188124975126, 2.58410224238832967050439894044, 3.29947709362510016722678506907, 4.05773382380719118038019486489, 5.16102648122241101840843222641, 5.89331222566998407995159637505, 7.21474648018584727348480981586, 7.51328561520881343456728042037, 8.130858904056281548866533661730