L(s) = 1 | + 2·3-s + 9-s + 2·11-s − 6·13-s − 17-s + 4·19-s + 4·23-s − 5·25-s − 4·27-s − 8·31-s + 4·33-s − 4·37-s − 12·39-s + 6·41-s + 8·43-s − 8·47-s − 7·49-s − 2·51-s + 10·53-s + 8·57-s + 12·61-s + 8·67-s + 8·69-s + 12·71-s + 2·73-s − 10·75-s − 4·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.242·17-s + 0.917·19-s + 0.834·23-s − 25-s − 0.769·27-s − 1.43·31-s + 0.696·33-s − 0.657·37-s − 1.92·39-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 49-s − 0.280·51-s + 1.37·53-s + 1.05·57-s + 1.53·61-s + 0.977·67-s + 0.963·69-s + 1.42·71-s + 0.234·73-s − 1.15·75-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.433630837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433630837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.37507819601213255380302574106, −12.31736659192486245303572672240, −11.24949424038558723722333629483, −9.716492599218371711354133604234, −9.181675232501275689341461374019, −7.915799644978553160956414624199, −7.04028809245781208083955490002, −5.27740412064621881845926355386, −3.69893087635466446387661413772, −2.33145296998291404024563148829,
2.33145296998291404024563148829, 3.69893087635466446387661413772, 5.27740412064621881845926355386, 7.04028809245781208083955490002, 7.915799644978553160956414624199, 9.181675232501275689341461374019, 9.716492599218371711354133604234, 11.24949424038558723722333629483, 12.31736659192486245303572672240, 13.37507819601213255380302574106