L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s − 4·5-s + 6·6-s − 4·7-s + 6·9-s + 8·10-s − 6·11-s − 6·12-s − 4·13-s + 8·14-s + 12·15-s − 4·16-s − 4·17-s − 12·18-s − 7·19-s − 8·20-s + 12·21-s + 12·22-s − 6·23-s + 11·25-s + 8·26-s − 9·27-s − 8·28-s − 6·29-s − 24·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s − 1.78·5-s + 2.44·6-s − 1.51·7-s + 2·9-s + 2.52·10-s − 1.80·11-s − 1.73·12-s − 1.10·13-s + 2.13·14-s + 3.09·15-s − 16-s − 0.970·17-s − 2.82·18-s − 1.60·19-s − 1.78·20-s + 2.61·21-s + 2.55·22-s − 1.25·23-s + 11/5·25-s + 1.56·26-s − 1.73·27-s − 1.51·28-s − 1.11·29-s − 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5077 | \( 1+O(T) \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.34281497953964814691434021056, −6.62250461340770678139848771792, −6.01192275298639519014642522249, −4.75443151596340586415163559397, −4.47055151331009795091782387950, −3.26244355597875746635580364386, −2.05247285847993976968922276314, 0, 0, 0,
2.05247285847993976968922276314, 3.26244355597875746635580364386, 4.47055151331009795091782387950, 4.75443151596340586415163559397, 6.01192275298639519014642522249, 6.62250461340770678139848771792, 7.34281497953964814691434021056