| L(s) = 1 | + 3-s + 7-s − 2·9-s − 5·11-s − 13-s − 3·17-s − 6·19-s + 21-s + 6·23-s − 5·27-s − 9·29-s − 5·33-s − 6·37-s − 39-s + 8·41-s − 6·43-s − 3·47-s + 49-s − 3·51-s + 12·53-s − 6·57-s + 8·59-s − 4·61-s − 2·63-s + 4·67-s + 6·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 0.727·17-s − 1.37·19-s + 0.218·21-s + 1.25·23-s − 0.962·27-s − 1.67·29-s − 0.870·33-s − 0.986·37-s − 0.160·39-s + 1.24·41-s − 0.914·43-s − 0.437·47-s + 1/7·49-s − 0.420·51-s + 1.64·53-s − 0.794·57-s + 1.04·59-s − 0.512·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−8.923791913883665586804923257643, −8.462219312931430130295138543437, −7.66737852056154649182867009604, −6.87549713237471057055540036166, −5.67002771047620563739767976211, −5.02967376320291988227199382319, −3.92102309478332339985108610603, −2.76252169464449428978311315426, −2.08175715199719083636321296262, 0,
2.08175715199719083636321296262, 2.76252169464449428978311315426, 3.92102309478332339985108610603, 5.02967376320291988227199382319, 5.67002771047620563739767976211, 6.87549713237471057055540036166, 7.66737852056154649182867009604, 8.462219312931430130295138543437, 8.923791913883665586804923257643
