| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s + 22-s + 24-s + 25-s − 26-s + 27-s − 28-s − 29-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s + 22-s + 24-s + 25-s − 26-s + 27-s − 28-s − 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.745833003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.745833003\) |
| \(L(1)\) |
\(\approx\) |
\(2.908921723\) |
| \(L(1)\) |
\(\approx\) |
\(2.908921723\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| 47 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 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
−21.67131293900945763812130659747, −20.596571102183441239901698368396, −20.08020150207028048408530525847, −19.46297576443421815593787265041, −18.601288948903272617485622385033, −17.379406665626773139610845261530, −16.62088832744908095003803736511, −15.76107601093799360937393153642, −14.97915573399433647882696673821, −14.18660656598997257415544189266, −13.749475276820025291588838030789, −12.88806411836543835617588256404, −12.42709605352602073663763159650, −11.23571760662968454427358333575, −10.08666725232798036238586597025, −9.53933159196399919972063241008, −8.79815283035585871875488928404, −7.267189128856644009605939767604, −6.90195102560315726304378105099, −5.93493284317087808744429326441, −4.968418335623515851100000547289, −3.91853183653632218864928451134, −3.153432804069686743342111036867, −2.31315821859996991079746213551, −1.53024267365364252532023398565,
1.53024267365364252532023398565, 2.31315821859996991079746213551, 3.153432804069686743342111036867, 3.91853183653632218864928451134, 4.968418335623515851100000547289, 5.93493284317087808744429326441, 6.90195102560315726304378105099, 7.267189128856644009605939767604, 8.79815283035585871875488928404, 9.53933159196399919972063241008, 10.08666725232798036238586597025, 11.23571760662968454427358333575, 12.42709605352602073663763159650, 12.88806411836543835617588256404, 13.749475276820025291588838030789, 14.18660656598997257415544189266, 14.97915573399433647882696673821, 15.76107601093799360937393153642, 16.62088832744908095003803736511, 17.379406665626773139610845261530, 18.601288948903272617485622385033, 19.46297576443421815593787265041, 20.08020150207028048408530525847, 20.596571102183441239901698368396, 21.67131293900945763812130659747
