L(s) = 1 | + 3-s + 4·5-s + 3·7-s − 2·9-s − 2·11-s + 13-s + 4·15-s + 3·17-s + 19-s + 3·21-s − 23-s + 11·25-s − 5·27-s + 5·29-s − 8·31-s − 2·33-s + 12·35-s + 2·37-s + 39-s − 8·41-s − 4·43-s − 8·45-s + 8·47-s + 2·49-s + 3·51-s + 53-s − 8·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1.13·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s + 1.03·15-s + 0.727·17-s + 0.229·19-s + 0.654·21-s − 0.208·23-s + 11/5·25-s − 0.962·27-s + 0.928·29-s − 1.43·31-s − 0.348·33-s + 2.02·35-s + 0.328·37-s + 0.160·39-s − 1.24·41-s − 0.609·43-s − 1.19·45-s + 1.16·47-s + 2/7·49-s + 0.420·51-s + 0.137·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.896470712\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.896470712\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.673084911618933118181680374348, −8.905991841422638352773015657402, −8.264845552131759886230905508728, −7.41068992206199799917476516665, −6.17646264461499634926610391599, −5.50702009116960948045704530045, −4.86338510531971803230901631456, −3.27846026315236304587310905813, −2.31224961625843070084602752549, −1.49207601742473373563863301755,
1.49207601742473373563863301755, 2.31224961625843070084602752549, 3.27846026315236304587310905813, 4.86338510531971803230901631456, 5.50702009116960948045704530045, 6.17646264461499634926610391599, 7.41068992206199799917476516665, 8.264845552131759886230905508728, 8.905991841422638352773015657402, 9.673084911618933118181680374348