L(s) = 1 | + 1.15·3-s − 4.09·5-s + 0.621·7-s − 1.66·9-s − 0.176·11-s − 6.62·13-s − 4.73·15-s − 4.87·17-s + 6.67·19-s + 0.718·21-s − 1.95·23-s + 11.7·25-s − 5.39·27-s − 4.79·29-s + 3.71·31-s − 0.204·33-s − 2.54·35-s + 6.22·37-s − 7.65·39-s + 1.84·41-s + 6.89·43-s + 6.81·45-s + 0.545·47-s − 6.61·49-s − 5.63·51-s − 4.30·53-s + 0.723·55-s + ⋯ |
L(s) = 1 | + 0.667·3-s − 1.83·5-s + 0.234·7-s − 0.554·9-s − 0.0532·11-s − 1.83·13-s − 1.22·15-s − 1.18·17-s + 1.53·19-s + 0.156·21-s − 0.406·23-s + 2.35·25-s − 1.03·27-s − 0.891·29-s + 0.666·31-s − 0.0355·33-s − 0.430·35-s + 1.02·37-s − 1.22·39-s + 0.288·41-s + 1.05·43-s + 1.01·45-s + 0.0795·47-s − 0.944·49-s − 0.789·51-s − 0.591·53-s + 0.0975·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9269796099\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9269796099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 - 0.621T + 7T^{2} \) |
| 11 | \( 1 + 0.176T + 11T^{2} \) |
| 13 | \( 1 + 6.62T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 19 | \( 1 - 6.67T + 19T^{2} \) |
| 23 | \( 1 + 1.95T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 - 3.71T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 - 0.545T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 - 6.86T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 18.1T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 8.30T + 97T^{2} \) |
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
−8.124617864639902889890675194746, −7.79101168139561346344531421637, −7.38728080890983022523867494313, −6.43513198237761692628426893130, −5.15981891989092126431418175173, −4.61148561737165257785538511987, −3.77621272111550057015793839375, −3.00404374352961898501404347763, −2.26966536817348726753482445218, −0.50312947777833023468855912453,
0.50312947777833023468855912453, 2.26966536817348726753482445218, 3.00404374352961898501404347763, 3.77621272111550057015793839375, 4.61148561737165257785538511987, 5.15981891989092126431418175173, 6.43513198237761692628426893130, 7.38728080890983022523867494313, 7.79101168139561346344531421637, 8.124617864639902889890675194746