L(s) = 1 | + 0.899·2-s − 1.19·4-s + 0.803·5-s − 7-s − 2.87·8-s + 0.723·10-s − 2.94·13-s − 0.899·14-s − 0.202·16-s + 3.30·17-s − 7.93·19-s − 0.956·20-s + 7.53·23-s − 4.35·25-s − 2.65·26-s + 1.19·28-s − 0.234·29-s + 3.98·31-s + 5.55·32-s + 2.97·34-s − 0.803·35-s − 0.536·37-s − 7.14·38-s − 2.30·40-s − 4.78·41-s − 3.16·43-s + 6.78·46-s + ⋯ |
L(s) = 1 | + 0.636·2-s − 0.595·4-s + 0.359·5-s − 0.377·7-s − 1.01·8-s + 0.228·10-s − 0.817·13-s − 0.240·14-s − 0.0506·16-s + 0.801·17-s − 1.82·19-s − 0.213·20-s + 1.57·23-s − 0.870·25-s − 0.520·26-s + 0.224·28-s − 0.0435·29-s + 0.715·31-s + 0.982·32-s + 0.509·34-s − 0.135·35-s − 0.0881·37-s − 1.15·38-s − 0.364·40-s − 0.747·41-s − 0.482·43-s + 0.999·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.646107871\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646107871\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.899T + 2T^{2} \) |
| 5 | \( 1 - 0.803T + 5T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 23 | \( 1 - 7.53T + 23T^{2} \) |
| 29 | \( 1 + 0.234T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 + 0.536T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 + 7.26T + 83T^{2} \) |
| 89 | \( 1 + 8.03T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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
−7.85109659143356968088712398090, −7.07655549586391300715568316111, −6.25050817531719786182475391540, −5.80214707975243313221222709845, −4.89037792176547737232216532182, −4.50605436195398061996041263350, −3.54263726012336775680175968939, −2.89813093033358409963091340650, −1.95701862334829435622944409839, −0.56673786017223428580767637651,
0.56673786017223428580767637651, 1.95701862334829435622944409839, 2.89813093033358409963091340650, 3.54263726012336775680175968939, 4.50605436195398061996041263350, 4.89037792176547737232216532182, 5.80214707975243313221222709845, 6.25050817531719786182475391540, 7.07655549586391300715568316111, 7.85109659143356968088712398090