L(s) = 1 | − 1.24·2-s − 0.445·4-s − 2.35·7-s + 3.04·8-s + 0.692·11-s − 2.89·13-s + 2.93·14-s − 2.91·16-s + 5.60·17-s + 2.04·19-s − 0.862·22-s + 0.533·23-s − 5·25-s + 3.60·26-s + 1.04·28-s + 1.60·29-s + 0.911·31-s − 2.46·32-s − 6.98·34-s + 0.542·37-s − 2.55·38-s − 8.39·41-s + 2.98·43-s − 0.307·44-s − 0.664·46-s − 6.27·47-s − 1.44·49-s + ⋯ |
L(s) = 1 | − 0.881·2-s − 0.222·4-s − 0.890·7-s + 1.07·8-s + 0.208·11-s − 0.801·13-s + 0.785·14-s − 0.727·16-s + 1.35·17-s + 0.470·19-s − 0.183·22-s + 0.111·23-s − 25-s + 0.706·26-s + 0.198·28-s + 0.297·29-s + 0.163·31-s − 0.436·32-s − 1.19·34-s + 0.0892·37-s − 0.414·38-s − 1.31·41-s + 0.455·43-s − 0.0464·44-s − 0.0980·46-s − 0.915·47-s − 0.206·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 0.692T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 - 0.533T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 - 0.911T + 31T^{2} \) |
| 37 | \( 1 - 0.542T + 37T^{2} \) |
| 41 | \( 1 + 8.39T + 41T^{2} \) |
| 43 | \( 1 - 2.98T + 43T^{2} \) |
| 47 | \( 1 + 6.27T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 0.417T + 61T^{2} \) |
| 67 | \( 1 + 8.11T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−9.328971878056703393591717727326, −8.505137847514147731934454599099, −7.65079581921701142254319029660, −7.06162858039986131026833990934, −5.93272534477260300850918642052, −5.02472191356584595001287841678, −3.93815737468169815065788003833, −2.92045889513514610390954544421, −1.41106791497134763498515720130, 0,
1.41106791497134763498515720130, 2.92045889513514610390954544421, 3.93815737468169815065788003833, 5.02472191356584595001287841678, 5.93272534477260300850918642052, 7.06162858039986131026833990934, 7.65079581921701142254319029660, 8.505137847514147731934454599099, 9.328971878056703393591717727326