L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s + 13-s + 15-s + 17-s + 21-s − 23-s + 25-s − 27-s + 29-s + 31-s − 33-s + 35-s + 37-s − 39-s − 41-s + 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s − 59-s + ⋯ |
L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s + 13-s + 15-s + 17-s + 21-s − 23-s + 25-s − 27-s + 29-s + 31-s − 33-s + 35-s + 37-s − 39-s − 41-s + 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s − 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6607783624\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6607783624\) |
\(L(1)\) |
\(\approx\) |
\(0.6981819238\) |
\(L(1)\) |
\(\approx\) |
\(0.6981819238\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.974782612223435936815828812713, −27.36394729141557122212844072540, −26.18920732272669110058801228255, −25.03570656031738438373345491671, −23.82656002911302603490190355199, −23.03546517339361911590564687320, −22.4831171975902518081338606882, −21.323986033541699543671381978759, −19.921211032466099821880850754791, −19.07930720260468266730379929029, −18.1480462056532279261661683663, −16.75391232872452868609638625265, −16.177410472479812603270356473786, −15.27043925225149444402190729235, −13.726439311685062036019092725554, −12.3641412344125729366071751428, −11.847943451353306640368770761690, −10.67872541432986864561167177667, −9.57362478597520850000394415294, −8.10096198733225825370167169296, −6.76191662004457664971702246735, −5.99381677708181103786151993262, −4.36625746846417773525563403107, −3.417274095235402145580073033205, −0.982619117315426971064820658161,
0.982619117315426971064820658161, 3.417274095235402145580073033205, 4.36625746846417773525563403107, 5.99381677708181103786151993262, 6.76191662004457664971702246735, 8.10096198733225825370167169296, 9.57362478597520850000394415294, 10.67872541432986864561167177667, 11.847943451353306640368770761690, 12.3641412344125729366071751428, 13.726439311685062036019092725554, 15.27043925225149444402190729235, 16.177410472479812603270356473786, 16.75391232872452868609638625265, 18.1480462056532279261661683663, 19.07930720260468266730379929029, 19.921211032466099821880850754791, 21.323986033541699543671381978759, 22.4831171975902518081338606882, 23.03546517339361911590564687320, 23.82656002911302603490190355199, 25.03570656031738438373345491671, 26.18920732272669110058801228255, 27.36394729141557122212844072540, 27.974782612223435936815828812713