| L(s) = 1 | − 2.66·2-s + 5.08·4-s + 3.46·7-s − 8.21·8-s − 3.43·11-s + 5.70·13-s − 9.23·14-s + 11.6·16-s + 1.89·17-s + 2.46·19-s + 9.13·22-s + 8.84·23-s − 15.1·26-s + 17.6·28-s + 8.98·29-s − 1.90·31-s − 14.7·32-s − 5.03·34-s + 5.09·37-s − 6.57·38-s + 6.35·41-s − 3.43·43-s − 17.4·44-s − 23.5·46-s + 8.61·47-s + 5.02·49-s + 29.0·52-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.54·4-s + 1.31·7-s − 2.90·8-s − 1.03·11-s + 1.58·13-s − 2.46·14-s + 2.92·16-s + 0.458·17-s + 0.566·19-s + 1.94·22-s + 1.84·23-s − 2.97·26-s + 3.33·28-s + 1.66·29-s − 0.342·31-s − 2.60·32-s − 0.863·34-s + 0.837·37-s − 1.06·38-s + 0.991·41-s − 0.523·43-s − 2.63·44-s − 3.47·46-s + 1.25·47-s + 0.718·49-s + 4.02·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.205312158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.205312158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 8.84T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 + 3.43T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 2.03T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 - 6.95T + 67T^{2} \) |
| 71 | \( 1 - 2.91T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 6.60T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 2.17T + 97T^{2} \) |
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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
−8.201590931792182726896065261343, −7.71449665175950713507805937963, −7.12037098265556745357804225202, −6.21277637928280832267622817112, −5.49441198113124862367775488758, −4.59835275883113257795201434556, −3.20920672122029436848582925096, −2.51142708551713243934303710408, −1.35929699446451392608734575075, −0.912425096168884309632086098525,
0.912425096168884309632086098525, 1.35929699446451392608734575075, 2.51142708551713243934303710408, 3.20920672122029436848582925096, 4.59835275883113257795201434556, 5.49441198113124862367775488758, 6.21277637928280832267622817112, 7.12037098265556745357804225202, 7.71449665175950713507805937963, 8.201590931792182726896065261343
