L(s) = 1 | − 3-s − 2·5-s + 9-s − 2·11-s + 2·15-s + 2·17-s − 2·23-s − 25-s − 27-s − 6·29-s + 4·31-s + 2·33-s − 6·37-s − 2·41-s − 2·45-s − 2·51-s + 6·53-s + 4·55-s + 12·59-s − 12·61-s − 12·67-s + 2·69-s + 10·71-s + 12·73-s + 75-s − 12·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.516·15-s + 0.485·17-s − 0.417·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.312·41-s − 0.298·45-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.56·59-s − 1.53·61-s − 1.46·67-s + 0.240·69-s + 1.18·71-s + 1.40·73-s + 0.115·75-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7595778354\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7595778354\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.74956152001126503915566771085, −7.07307427216897839519049786071, −6.36489276263918734001714029212, −5.52669987985987992536944439284, −5.06978801641505802327886357374, −4.11900015768954548325726703631, −3.63913502580163940020275723131, −2.66297315279224472319784967994, −1.61795258338161771411997168217, −0.43046359077281671538431978521,
0.43046359077281671538431978521, 1.61795258338161771411997168217, 2.66297315279224472319784967994, 3.63913502580163940020275723131, 4.11900015768954548325726703631, 5.06978801641505802327886357374, 5.52669987985987992536944439284, 6.36489276263918734001714029212, 7.07307427216897839519049786071, 7.74956152001126503915566771085