L(s) = 1 | + (−0.985 + 0.169i)4-s + (1.71 + 0.776i)7-s + (−0.481 − 0.147i)13-s + (0.942 − 0.333i)16-s + (−0.386 + 0.439i)19-s + (0.873 + 0.487i)25-s + (−1.82 − 0.475i)28-s + (−0.183 − 0.175i)31-s + (−0.538 + 1.33i)37-s + (1.15 − 0.644i)43-s + (1.68 + 1.91i)49-s + (0.499 + 0.0640i)52-s + (−0.787 − 0.241i)61-s + (−0.873 + 0.487i)64-s + (0.301 + 1.76i)67-s + ⋯ |
L(s) = 1 | + (−0.985 + 0.169i)4-s + (1.71 + 0.776i)7-s + (−0.481 − 0.147i)13-s + (0.942 − 0.333i)16-s + (−0.386 + 0.439i)19-s + (0.873 + 0.487i)25-s + (−1.82 − 0.475i)28-s + (−0.183 − 0.175i)31-s + (−0.538 + 1.33i)37-s + (1.15 − 0.644i)43-s + (1.68 + 1.91i)49-s + (0.499 + 0.0640i)52-s + (−0.787 − 0.241i)61-s + (−0.873 + 0.487i)64-s + (0.301 + 1.76i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.086177724\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.086177724\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 + (0.873 + 0.487i)T \) |
good | 2 | \( 1 + (0.985 - 0.169i)T^{2} \) |
| 5 | \( 1 + (-0.873 - 0.487i)T^{2} \) |
| 7 | \( 1 + (-1.71 - 0.776i)T + (0.660 + 0.750i)T^{2} \) |
| 11 | \( 1 + (0.721 - 0.691i)T^{2} \) |
| 13 | \( 1 + (0.481 + 0.147i)T + (0.828 + 0.559i)T^{2} \) |
| 17 | \( 1 + (-0.721 + 0.691i)T^{2} \) |
| 19 | \( 1 + (0.386 - 0.439i)T + (-0.127 - 0.991i)T^{2} \) |
| 23 | \( 1 + (-0.873 - 0.487i)T^{2} \) |
| 29 | \( 1 + (0.778 - 0.628i)T^{2} \) |
| 31 | \( 1 + (0.183 + 0.175i)T + (0.0424 + 0.999i)T^{2} \) |
| 37 | \( 1 + (0.538 - 1.33i)T + (-0.721 - 0.691i)T^{2} \) |
| 41 | \( 1 + (-0.292 - 0.956i)T^{2} \) |
| 43 | \( 1 + (-1.15 + 0.644i)T + (0.524 - 0.851i)T^{2} \) |
| 47 | \( 1 + (-0.127 + 0.991i)T^{2} \) |
| 53 | \( 1 + (-0.911 + 0.411i)T^{2} \) |
| 59 | \( 1 + (0.594 + 0.803i)T^{2} \) |
| 61 | \( 1 + (0.787 + 0.241i)T + (0.828 + 0.559i)T^{2} \) |
| 67 | \( 1 + (-0.301 - 1.76i)T + (-0.942 + 0.333i)T^{2} \) |
| 71 | \( 1 + (0.594 + 0.803i)T^{2} \) |
| 73 | \( 1 + (-0.250 + 1.95i)T + (-0.967 - 0.251i)T^{2} \) |
| 79 | \( 1 + (-0.784 - 0.579i)T + (0.292 + 0.956i)T^{2} \) |
| 83 | \( 1 + (0.524 + 0.851i)T^{2} \) |
| 89 | \( 1 + (0.873 - 0.487i)T^{2} \) |
| 97 | \( 1 + (-0.405 - 1.55i)T + (-0.873 + 0.487i)T^{2} \) |
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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
−9.152717047070456443626789064207, −8.695333115349240942020146745177, −8.005939727947818898594050433137, −7.40374579658185585983062696709, −6.06675928361181729444281317135, −5.11851132196001626207891362755, −4.82610725196068100185753377325, −3.80471037783736353580242416576, −2.55132967108389766066190531976, −1.38262654032254869950646932586,
0.957921428897014067004852950262, 2.16212764764612032252591686251, 3.68812870085335233444155623771, 4.62250007408118678806876260914, 4.87093181876810966611832723051, 5.90546038506127819978667515643, 7.14444432690146034917723229995, 7.75312811071546854204355316427, 8.532996626885514690358649934122, 9.082732791285321203186462669628