| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s + 15-s + 16-s − 18-s − 19-s + 20-s − 24-s + 25-s + 27-s − 30-s − 32-s + 36-s + 38-s − 40-s + 45-s + 48-s − 49-s − 50-s + 2·53-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s + 15-s + 16-s − 18-s − 19-s + 20-s − 24-s + 25-s + 27-s − 30-s − 32-s + 36-s + 38-s − 40-s + 45-s + 48-s − 49-s − 50-s + 2·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.329796046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.329796046\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−9.084131124151367417930992213759, −8.639436023949171477147443277439, −7.87345556479182729543855813325, −7.00193213907656420246777215187, −6.39581896686075972259297591229, −5.44882079571275726599661901873, −4.20364094806101575200197235135, −3.01642336475461301738097983419, −2.26894086475307031536946575594, −1.41470701784532426951276863726,
1.41470701784532426951276863726, 2.26894086475307031536946575594, 3.01642336475461301738097983419, 4.20364094806101575200197235135, 5.44882079571275726599661901873, 6.39581896686075972259297591229, 7.00193213907656420246777215187, 7.87345556479182729543855813325, 8.639436023949171477147443277439, 9.084131124151367417930992213759
