L(s) = 1 | − 2·5-s − 2·11-s − 13-s + 19-s − 25-s − 4·29-s + 9·31-s + 3·37-s − 10·41-s − 5·43-s + 6·47-s − 12·53-s + 4·55-s + 12·59-s − 10·61-s + 2·65-s + 5·67-s − 6·71-s + 3·73-s + 79-s − 6·83-s + 16·89-s − 2·95-s + 6·97-s + 2·101-s − 7·103-s − 8·107-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.603·11-s − 0.277·13-s + 0.229·19-s − 1/5·25-s − 0.742·29-s + 1.61·31-s + 0.493·37-s − 1.56·41-s − 0.762·43-s + 0.875·47-s − 1.64·53-s + 0.539·55-s + 1.56·59-s − 1.28·61-s + 0.248·65-s + 0.610·67-s − 0.712·71-s + 0.351·73-s + 0.112·79-s − 0.658·83-s + 1.69·89-s − 0.205·95-s + 0.609·97-s + 0.199·101-s − 0.689·103-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.108543546\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.108543546\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.017589702404721192614417980051, −7.33049509805256051250877801181, −6.65122105587384884743109466965, −5.81876503333370748710572863449, −4.99958062539043986465506141851, −4.39630329525425863537663342497, −3.53648765279739585071717659976, −2.85265542341552762183879045973, −1.82115415109922650847218607880, −0.52283799263477174247860807149,
0.52283799263477174247860807149, 1.82115415109922650847218607880, 2.85265542341552762183879045973, 3.53648765279739585071717659976, 4.39630329525425863537663342497, 4.99958062539043986465506141851, 5.81876503333370748710572863449, 6.65122105587384884743109466965, 7.33049509805256051250877801181, 8.017589702404721192614417980051