| L(s) = 1 | − 0.104·2-s − 1.98·4-s + 0.431·5-s + 2.76·7-s + 0.415·8-s − 0.0449·10-s + 1.08·11-s − 2.81·13-s − 0.288·14-s + 3.93·16-s − 5.82·17-s − 7.25·19-s − 0.859·20-s − 0.113·22-s + 3.48·23-s − 4.81·25-s + 0.292·26-s − 5.50·28-s + 5.03·29-s − 2.02·31-s − 1.24·32-s + 0.606·34-s + 1.19·35-s + 2.08·37-s + 0.755·38-s + 0.179·40-s + 5.40·41-s + ⋯ |
| L(s) = 1 | − 0.0736·2-s − 0.994·4-s + 0.193·5-s + 1.04·7-s + 0.146·8-s − 0.0142·10-s + 0.328·11-s − 0.779·13-s − 0.0769·14-s + 0.983·16-s − 1.41·17-s − 1.66·19-s − 0.192·20-s − 0.0241·22-s + 0.727·23-s − 0.962·25-s + 0.0574·26-s − 1.03·28-s + 0.935·29-s − 0.363·31-s − 0.219·32-s + 0.104·34-s + 0.202·35-s + 0.342·37-s + 0.122·38-s + 0.0283·40-s + 0.843·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 + 0.104T + 2T^{2} \) |
| 5 | \( 1 - 0.431T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 - 1.08T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 - 5.40T + 41T^{2} \) |
| 43 | \( 1 + 5.91T + 43T^{2} \) |
| 47 | \( 1 - 0.795T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 - 9.03T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 79 | \( 1 + 0.409T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.672040726079254107717999354862, −8.166236732672892918534618852460, −7.21718748685566793565173447621, −6.31064611626787532151448503121, −5.33078058536277384000252490006, −4.48839934083428248930867094401, −4.16727604521443044754487572500, −2.60301233676385610031142503513, −1.56666171496230014263661095266, 0,
1.56666171496230014263661095266, 2.60301233676385610031142503513, 4.16727604521443044754487572500, 4.48839934083428248930867094401, 5.33078058536277384000252490006, 6.31064611626787532151448503121, 7.21718748685566793565173447621, 8.166236732672892918534618852460, 8.672040726079254107717999354862
