L(s) = 1 | + 5-s + 7-s − 3·9-s + 6·11-s − 6·13-s + 7·17-s + 2·19-s + 23-s + 25-s + 5·29-s − 31-s + 35-s + 5·37-s − 7·41-s + 8·43-s − 3·45-s − 8·47-s − 6·49-s − 3·53-s + 6·55-s + 13·59-s + 8·61-s − 3·63-s − 6·65-s − 9·67-s − 7·71-s − 2·73-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s + 1.80·11-s − 1.66·13-s + 1.69·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.928·29-s − 0.179·31-s + 0.169·35-s + 0.821·37-s − 1.09·41-s + 1.21·43-s − 0.447·45-s − 1.16·47-s − 6/7·49-s − 0.412·53-s + 0.809·55-s + 1.69·59-s + 1.02·61-s − 0.377·63-s − 0.744·65-s − 1.09·67-s − 0.830·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.451435034\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.451435034\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.924693249171455614929089331959, −7.17767069158269665114114042451, −6.52265524987877369624607953428, −5.74594888450684997900703952276, −5.16755127869188188210521918313, −4.43292359076631388397587095637, −3.41033058466000776145174841299, −2.77587045885060386364298128090, −1.73995837676275608339674615270, −0.820828434112242215940875789316,
0.820828434112242215940875789316, 1.73995837676275608339674615270, 2.77587045885060386364298128090, 3.41033058466000776145174841299, 4.43292359076631388397587095637, 5.16755127869188188210521918313, 5.74594888450684997900703952276, 6.52265524987877369624607953428, 7.17767069158269665114114042451, 7.924693249171455614929089331959