L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 2·11-s + 12-s + 3·13-s + 14-s + 16-s + 4·17-s + 18-s + 21-s − 2·22-s + 4·23-s + 24-s − 5·25-s + 3·26-s + 27-s + 28-s + 3·31-s + 32-s − 2·33-s + 4·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.218·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s − 25-s + 0.588·26-s + 0.192·27-s + 0.188·28-s + 0.538·31-s + 0.176·32-s − 0.348·33-s + 0.685·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.799455844\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.799455844\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.925966099863985529054739651698, −8.180578212475829505283816182977, −7.59187105262908473388016033500, −6.71460371174360847262151229869, −5.76177764095449448246208886611, −5.08258719095464042056098916585, −4.08698162864034153602834325880, −3.32245950530557880865147334790, −2.41032538782090925484227896417, −1.24755944995887851640988741563,
1.24755944995887851640988741563, 2.41032538782090925484227896417, 3.32245950530557880865147334790, 4.08698162864034153602834325880, 5.08258719095464042056098916585, 5.76177764095449448246208886611, 6.71460371174360847262151229869, 7.59187105262908473388016033500, 8.180578212475829505283816182977, 8.925966099863985529054739651698