L(s) = 1 | − 5·7-s + 7·13-s − 19-s − 5·25-s + 4·31-s + 37-s + 8·43-s + 18·49-s + 13·61-s + 11·67-s + 17·73-s + 13·79-s − 35·91-s + 5·97-s + 7·103-s − 2·109-s + ⋯ |
L(s) = 1 | − 1.88·7-s + 1.94·13-s − 0.229·19-s − 25-s + 0.718·31-s + 0.164·37-s + 1.21·43-s + 18/7·49-s + 1.66·61-s + 1.34·67-s + 1.98·73-s + 1.46·79-s − 3.66·91-s + 0.507·97-s + 0.689·103-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.363034090\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.363034090\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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
−9.353701574803802413804076848425, −8.650409645340224009044341041842, −7.78450025598181104413322457077, −6.59048869509056337119790477940, −6.31866267091677471011901067110, −5.48216093066766077645578315510, −3.94546281439316433923646778556, −3.55340608613286864344461522402, −2.42247921633551671696045563684, −0.798653294514055210127523925209,
0.798653294514055210127523925209, 2.42247921633551671696045563684, 3.55340608613286864344461522402, 3.94546281439316433923646778556, 5.48216093066766077645578315510, 6.31866267091677471011901067110, 6.59048869509056337119790477940, 7.78450025598181104413322457077, 8.650409645340224009044341041842, 9.353701574803802413804076848425