L(s) = 1 | + 2.39·2-s − 3-s + 3.71·4-s − 2.58·5-s − 2.39·6-s + 7-s + 4.09·8-s + 9-s − 6.19·10-s − 3.71·12-s − 6.78·13-s + 2.39·14-s + 2.58·15-s + 2.36·16-s + 3.14·17-s + 2.39·18-s + 1.04·19-s − 9.61·20-s − 21-s − 6.52·23-s − 4.09·24-s + 1.70·25-s − 16.2·26-s − 27-s + 3.71·28-s + 0.607·29-s + 6.19·30-s + ⋯ |
L(s) = 1 | + 1.69·2-s − 0.577·3-s + 1.85·4-s − 1.15·5-s − 0.975·6-s + 0.377·7-s + 1.44·8-s + 0.333·9-s − 1.95·10-s − 1.07·12-s − 1.88·13-s + 0.638·14-s + 0.668·15-s + 0.590·16-s + 0.762·17-s + 0.563·18-s + 0.239·19-s − 2.15·20-s − 0.218·21-s − 1.36·23-s − 0.835·24-s + 0.341·25-s − 3.17·26-s − 0.192·27-s + 0.701·28-s + 0.112·29-s + 1.13·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 17 | \( 1 - 3.14T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 - 0.607T + 29T^{2} \) |
| 31 | \( 1 + 8.83T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 1.89T + 53T^{2} \) |
| 59 | \( 1 + 0.174T + 59T^{2} \) |
| 61 | \( 1 + 8.13T + 61T^{2} \) |
| 67 | \( 1 + 7.33T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 + 0.589T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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
−8.043996957413836265728153860486, −7.49810974215811869086182876393, −6.92150887111400690683042950591, −5.86193134440943465279899268867, −5.23284313817329818224790798229, −4.53631469287683408170350038017, −3.93224392094103693532682784349, −3.03562273768791545050073872595, −1.93606233753585539488677769887, 0,
1.93606233753585539488677769887, 3.03562273768791545050073872595, 3.93224392094103693532682784349, 4.53631469287683408170350038017, 5.23284313817329818224790798229, 5.86193134440943465279899268867, 6.92150887111400690683042950591, 7.49810974215811869086182876393, 8.043996957413836265728153860486