L(s) = 1 | − 3-s − 5-s + 5·7-s + 9-s + 11-s + 13-s + 15-s + 3·17-s − 6·19-s − 5·21-s + 3·23-s + 25-s − 27-s + 4·29-s − 33-s − 5·35-s − 5·37-s − 39-s + 11·41-s + 6·43-s − 45-s + 18·49-s − 3·51-s − 9·53-s − 55-s + 6·57-s + 12·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.88·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 1.37·19-s − 1.09·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.174·33-s − 0.845·35-s − 0.821·37-s − 0.160·39-s + 1.71·41-s + 0.914·43-s − 0.149·45-s + 18/7·49-s − 0.420·51-s − 1.23·53-s − 0.134·55-s + 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.091059467\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.091059467\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.182259601808666893879113547520, −7.34840890241038769346249557930, −6.72310201603876121072382624247, −5.75740861302183262574293091424, −5.19141471269431269081981543735, −4.38279570716960452621577358455, −3.99963770775305372187722287929, −2.65042064379583256167307175530, −1.64069843773118666584792468096, −0.841040148618903126263745758406,
0.841040148618903126263745758406, 1.64069843773118666584792468096, 2.65042064379583256167307175530, 3.99963770775305372187722287929, 4.38279570716960452621577358455, 5.19141471269431269081981543735, 5.75740861302183262574293091424, 6.72310201603876121072382624247, 7.34840890241038769346249557930, 8.182259601808666893879113547520