L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 3·13-s + 15-s − 3·19-s + 21-s + 8·23-s − 4·25-s − 27-s + 7·29-s − 2·31-s + 35-s + 5·37-s + 3·39-s + 4·41-s + 4·43-s − 45-s − 3·47-s + 49-s − 6·53-s + 3·57-s − 3·59-s − 10·61-s − 63-s + 3·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.832·13-s + 0.258·15-s − 0.688·19-s + 0.218·21-s + 1.66·23-s − 4/5·25-s − 0.192·27-s + 1.29·29-s − 0.359·31-s + 0.169·35-s + 0.821·37-s + 0.480·39-s + 0.624·41-s + 0.609·43-s − 0.149·45-s − 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.397·57-s − 0.390·59-s − 1.28·61-s − 0.125·63-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9712400823\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9712400823\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.76008748309247, −15.13721634846610, −14.64934250657320, −14.07315479288349, −13.29942034857505, −12.79709924777097, −12.38867235511370, −11.82966481789378, −11.18525487903766, −10.79202146919236, −10.13937097695019, −9.506903065092915, −9.060838151983800, −8.225074227934926, −7.666481891407560, −7.057755922946065, −6.506488173498817, −5.913046940040322, −5.127585040935031, −4.576433867003169, −4.002072378052569, −3.061374622851916, −2.493048949910116, −1.386287901808369, −0.4378127256134325,
0.4378127256134325, 1.386287901808369, 2.493048949910116, 3.061374622851916, 4.002072378052569, 4.576433867003169, 5.127585040935031, 5.913046940040322, 6.506488173498817, 7.057755922946065, 7.666481891407560, 8.225074227934926, 9.060838151983800, 9.506903065092915, 10.13937097695019, 10.79202146919236, 11.18525487903766, 11.82966481789378, 12.38867235511370, 12.79709924777097, 13.29942034857505, 14.07315479288349, 14.64934250657320, 15.13721634846610, 15.76008748309247