| L(s) = 1 | − 2.58·3-s − 3.78·5-s + 3.69·9-s − 3.59·11-s − 3.80·13-s + 9.79·15-s − 4.90·17-s − 0.726·19-s + 2.67·23-s + 9.32·25-s − 1.80·27-s − 1.31·29-s − 4.60·31-s + 9.30·33-s + 6.36·37-s + 9.84·39-s − 41-s − 6.79·43-s − 13.9·45-s + 9.58·47-s + 12.6·51-s + 3.00·53-s + 13.6·55-s + 1.88·57-s − 4.98·59-s + 2.28·61-s + 14.4·65-s + ⋯ |
| L(s) = 1 | − 1.49·3-s − 1.69·5-s + 1.23·9-s − 1.08·11-s − 1.05·13-s + 2.52·15-s − 1.18·17-s − 0.166·19-s + 0.558·23-s + 1.86·25-s − 0.347·27-s − 0.244·29-s − 0.826·31-s + 1.62·33-s + 1.04·37-s + 1.57·39-s − 0.156·41-s − 1.03·43-s − 2.08·45-s + 1.39·47-s + 1.77·51-s + 0.412·53-s + 1.83·55-s + 0.249·57-s − 0.649·59-s + 0.292·61-s + 1.78·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 + 3.78T + 5T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 + 0.726T + 19T^{2} \) |
| 23 | \( 1 - 2.67T + 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 - 6.36T + 37T^{2} \) |
| 43 | \( 1 + 6.79T + 43T^{2} \) |
| 47 | \( 1 - 9.58T + 47T^{2} \) |
| 53 | \( 1 - 3.00T + 53T^{2} \) |
| 59 | \( 1 + 4.98T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 + 1.69T + 71T^{2} \) |
| 73 | \( 1 + 0.259T + 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 1.39T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33707709849567644263137066230, −6.94664864030827511288806743938, −6.08629068729075818516543602205, −5.22421145465675632820159210191, −4.73451693403289904368383861771, −4.21412669024391490993631415883, −3.21560388623510187049615496398, −2.23688288488944499148457754468, −0.65409146398852029129338771939, 0,
0.65409146398852029129338771939, 2.23688288488944499148457754468, 3.21560388623510187049615496398, 4.21412669024391490993631415883, 4.73451693403289904368383861771, 5.22421145465675632820159210191, 6.08629068729075818516543602205, 6.94664864030827511288806743938, 7.33707709849567644263137066230
