| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s − 7-s + 9-s + 2·10-s + 2·12-s − 13-s + 2·14-s − 15-s − 4·16-s + 17-s − 2·18-s + 6·19-s − 2·20-s − 21-s + 25-s + 2·26-s + 27-s − 2·28-s + 2·30-s + 4·31-s + 8·32-s − 2·34-s + 35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s + 0.577·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s − 16-s + 0.242·17-s − 0.471·18-s + 1.37·19-s − 0.447·20-s − 0.218·21-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.377·28-s + 0.365·30-s + 0.718·31-s + 1.41·32-s − 0.342·34-s + 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.476649539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476649539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.00660316816507, −12.37426306417442, −11.98874484742102, −11.50334334689840, −10.89119613954344, −10.61762083183076, −9.948860046495738, −9.609367241251526, −9.184016208801270, −8.984876459154625, −8.064473127664884, −7.838317707252734, −7.694373811185871, −6.896471610152941, −6.616167826365448, −5.890303982113853, −5.216892289620578, −4.598880690833465, −4.116875005744563, −3.405912309483196, −2.885881306594113, −2.385802738648280, −1.631196013823559, −0.9588908746569442, −0.5142639996983345,
0.5142639996983345, 0.9588908746569442, 1.631196013823559, 2.385802738648280, 2.885881306594113, 3.405912309483196, 4.116875005744563, 4.598880690833465, 5.216892289620578, 5.890303982113853, 6.616167826365448, 6.896471610152941, 7.694373811185871, 7.838317707252734, 8.064473127664884, 8.984876459154625, 9.184016208801270, 9.609367241251526, 9.948860046495738, 10.61762083183076, 10.89119613954344, 11.50334334689840, 11.98874484742102, 12.37426306417442, 13.00660316816507
