L(s) = 1 | − 3-s + 2·7-s + 9-s − 13-s + 4·17-s + 6·19-s − 2·21-s − 6·23-s − 27-s + 4·29-s + 8·31-s + 6·37-s + 39-s + 6·41-s − 4·43-s − 8·47-s − 3·49-s − 4·51-s − 2·53-s − 6·57-s − 2·61-s + 2·63-s + 4·67-s + 6·69-s − 8·71-s + 16·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.970·17-s + 1.37·19-s − 0.436·21-s − 1.25·23-s − 0.192·27-s + 0.742·29-s + 1.43·31-s + 0.986·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.794·57-s − 0.256·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.949·71-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.978213471\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.978213471\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.986622336300113308442054608512, −7.22795569086376819706411624696, −6.33949481008680480500487088605, −5.79600071764111841459942651372, −4.97946564076675613100018722733, −4.55018293538489446400556372225, −3.54791455598267040520875396042, −2.68727288426056290994599474575, −1.58864990401304168322356395338, −0.77106860614917570896246748945,
0.77106860614917570896246748945, 1.58864990401304168322356395338, 2.68727288426056290994599474575, 3.54791455598267040520875396042, 4.55018293538489446400556372225, 4.97946564076675613100018722733, 5.79600071764111841459942651372, 6.33949481008680480500487088605, 7.22795569086376819706411624696, 7.986622336300113308442054608512