L(s) = 1 | − 5-s + 3.74·7-s − 2.54·11-s + 13-s + 1.74·17-s − 2.29·19-s + 1.74·23-s + 25-s − 4.29·29-s + 2·31-s − 3.74·35-s + 8.03·37-s − 2.94·41-s + 7.49·43-s + 3.49·47-s + 7.03·49-s + 2.54·53-s + 2.54·55-s + 8.58·59-s − 4.03·61-s − 65-s − 1.49·67-s − 13.5·71-s + 9.78·73-s − 9.52·77-s + 8.94·79-s − 1.74·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.41·7-s − 0.766·11-s + 0.277·13-s + 0.423·17-s − 0.525·19-s + 0.364·23-s + 0.200·25-s − 0.796·29-s + 0.359·31-s − 0.633·35-s + 1.32·37-s − 0.460·41-s + 1.14·43-s + 0.509·47-s + 1.00·49-s + 0.349·53-s + 0.342·55-s + 1.11·59-s − 0.516·61-s − 0.124·65-s − 0.182·67-s − 1.60·71-s + 1.14·73-s − 1.08·77-s + 1.00·79-s − 0.189·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.043300398\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.043300398\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.03T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 - 7.49T + 43T^{2} \) |
| 47 | \( 1 - 3.49T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 - 8.58T + 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 9.78T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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
−8.139156487262176017171779110078, −7.75637092125941982544463909835, −7.06114372012639516551610402869, −5.98952739804179716483219474021, −5.30604815463230189791869732069, −4.58619658493030829889687381405, −3.92149068981492547736017743118, −2.81423144935841882497111775892, −1.91884298169708411739680759940, −0.811428288041246239039583187923,
0.811428288041246239039583187923, 1.91884298169708411739680759940, 2.81423144935841882497111775892, 3.92149068981492547736017743118, 4.58619658493030829889687381405, 5.30604815463230189791869732069, 5.98952739804179716483219474021, 7.06114372012639516551610402869, 7.75637092125941982544463909835, 8.139156487262176017171779110078