L(s) = 1 | − 2.63·2-s − 3-s + 4.92·4-s + 5-s + 2.63·6-s + 4.16·7-s − 7.69·8-s + 9-s − 2.63·10-s − 4.92·12-s − 5.26·13-s − 10.9·14-s − 15-s + 10.4·16-s − 4.23·17-s − 2.63·18-s + 2.50·19-s + 4.92·20-s − 4.16·21-s + 5.48·23-s + 7.69·24-s + 25-s + 13.8·26-s − 27-s + 20.5·28-s − 5.26·29-s + 2.63·30-s + ⋯ |
L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.46·4-s + 0.447·5-s + 1.07·6-s + 1.57·7-s − 2.72·8-s + 0.333·9-s − 0.832·10-s − 1.42·12-s − 1.45·13-s − 2.93·14-s − 0.258·15-s + 2.60·16-s − 1.02·17-s − 0.620·18-s + 0.574·19-s + 1.10·20-s − 0.909·21-s + 1.14·23-s + 1.57·24-s + 0.200·25-s + 2.71·26-s − 0.192·27-s + 3.87·28-s − 0.977·29-s + 0.480·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7253878749\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7253878749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 + 5.26T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4.48T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 + 3.63T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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
−9.245989143023734165906564912397, −8.608121810429003359026359621248, −7.67448216439699323302855970509, −7.27770942938493002582391848150, −6.40710560035774514124569498517, −5.32630889446843402776215303962, −4.60126583134952915227422915190, −2.62591865621093436354674313949, −1.86176657611354719519191783647, −0.803600380337510547049285616570,
0.803600380337510547049285616570, 1.86176657611354719519191783647, 2.62591865621093436354674313949, 4.60126583134952915227422915190, 5.32630889446843402776215303962, 6.40710560035774514124569498517, 7.27770942938493002582391848150, 7.67448216439699323302855970509, 8.608121810429003359026359621248, 9.245989143023734165906564912397