L(s) = 1 | + 2.05·2-s + 2.20·4-s + 1.27·7-s + 0.428·8-s + 3.17·11-s − 3.19·13-s + 2.62·14-s − 3.53·16-s + 1.35·17-s + 1.53·19-s + 6.51·22-s + 2.39·23-s − 6.54·26-s + 2.82·28-s + 6.80·29-s − 7.49·31-s − 8.11·32-s + 2.78·34-s + 5.24·37-s + 3.15·38-s + 6.41·41-s + 11.1·43-s + 7.01·44-s + 4.90·46-s + 12.4·47-s − 5.36·49-s − 7.05·52-s + ⋯ |
L(s) = 1 | + 1.45·2-s + 1.10·4-s + 0.483·7-s + 0.151·8-s + 0.957·11-s − 0.885·13-s + 0.701·14-s − 0.884·16-s + 0.329·17-s + 0.352·19-s + 1.38·22-s + 0.498·23-s − 1.28·26-s + 0.534·28-s + 1.26·29-s − 1.34·31-s − 1.43·32-s + 0.477·34-s + 0.861·37-s + 0.511·38-s + 1.00·41-s + 1.69·43-s + 1.05·44-s + 0.723·46-s + 1.81·47-s − 0.766·49-s − 0.977·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.890377646\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.890377646\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 7.49T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 - 6.41T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 0.474T + 53T^{2} \) |
| 59 | \( 1 - 8.04T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 6.45T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 1.77T + 73T^{2} \) |
| 79 | \( 1 + 0.781T + 79T^{2} \) |
| 83 | \( 1 + 1.43T + 83T^{2} \) |
| 89 | \( 1 - 7.89T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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
−7.88850181150866447026671538588, −7.20473725483252163300476437344, −6.58390442811291826337033328778, −5.70949267260511921009464130511, −5.25652861732830132437481567851, −4.35206960015091249817375700578, −3.98057508729505219830736436669, −2.93742235921106794461800941600, −2.26544532593130262995586715319, −0.978794356435778550886127273340,
0.978794356435778550886127273340, 2.26544532593130262995586715319, 2.93742235921106794461800941600, 3.98057508729505219830736436669, 4.35206960015091249817375700578, 5.25652861732830132437481567851, 5.70949267260511921009464130511, 6.58390442811291826337033328778, 7.20473725483252163300476437344, 7.88850181150866447026671538588