L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s + 9-s + 3·10-s − 4·11-s − 12-s − 14-s + 3·15-s + 16-s − 5·17-s − 18-s − 4·19-s − 3·20-s − 21-s + 4·22-s − 4·23-s + 24-s + 4·25-s − 27-s + 28-s − 9·29-s − 3·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.20·11-s − 0.288·12-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s − 0.218·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s + 4/5·25-s − 0.192·27-s + 0.188·28-s − 1.67·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.30786995059701323743155440462, −6.82449645497703084309440701731, −5.86536548328047928195824579804, −5.13270050660633047639794222892, −4.30187398269868867197594753414, −3.70002176190387304547844441984, −2.53374514055557526982099808168, −1.68592455953007718727796239134, 0, 0,
1.68592455953007718727796239134, 2.53374514055557526982099808168, 3.70002176190387304547844441984, 4.30187398269868867197594753414, 5.13270050660633047639794222892, 5.86536548328047928195824579804, 6.82449645497703084309440701731, 7.30786995059701323743155440462