| L(s) = 1 | + 3.40·7-s + 4.40·11-s − 2.06·13-s − 1.40·17-s − 6.35·19-s + 0.107·23-s − 9.08·29-s + 3.06·31-s − 1.95·37-s + 8.69·41-s + 7.12·43-s − 7.48·47-s + 4.57·49-s + 13.0·53-s + 13.1·59-s − 1.72·61-s + 12.3·67-s + 7.50·71-s + 5.42·73-s + 14.9·77-s + 9.43·79-s + 8.97·83-s − 4.01·89-s − 7.01·91-s − 2.17·97-s − 2.97·101-s − 6.63·103-s + ⋯ |
| L(s) = 1 | + 1.28·7-s + 1.32·11-s − 0.572·13-s − 0.339·17-s − 1.45·19-s + 0.0224·23-s − 1.68·29-s + 0.550·31-s − 0.321·37-s + 1.35·41-s + 1.08·43-s − 1.09·47-s + 0.653·49-s + 1.79·53-s + 1.71·59-s − 0.220·61-s + 1.51·67-s + 0.891·71-s + 0.634·73-s + 1.70·77-s + 1.06·79-s + 0.985·83-s − 0.425·89-s − 0.735·91-s − 0.220·97-s − 0.295·101-s − 0.653·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.472621210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.472621210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.40T + 7T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 - 0.107T + 23T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 - 5.42T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 - 8.97T + 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 + 2.17T + 97T^{2} \) |
| 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
−7.930528486365264273972457702085, −7.07747688590744331953730316536, −6.55961931324016457299391795706, −5.69066001353148556149577515316, −4.99788321742638863688452636472, −4.16761463009619293156030414958, −3.83483003996397274541603699765, −2.37166354133009151265083766242, −1.88299936515585409787180133940, −0.792409258528777316446321963421,
0.792409258528777316446321963421, 1.88299936515585409787180133940, 2.37166354133009151265083766242, 3.83483003996397274541603699765, 4.16761463009619293156030414958, 4.99788321742638863688452636472, 5.69066001353148556149577515316, 6.55961931324016457299391795706, 7.07747688590744331953730316536, 7.930528486365264273972457702085
