L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s − 2·9-s − 3·11-s − 2·12-s + 5·13-s − 15-s + 4·16-s + 3·17-s + 2·19-s + 2·20-s + 21-s − 6·23-s + 25-s − 5·27-s − 2·28-s + 3·29-s − 4·31-s − 3·33-s − 35-s + 4·36-s + 2·37-s + 5·39-s − 12·41-s − 10·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.727·17-s + 0.458·19-s + 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.377·28-s + 0.557·29-s − 0.718·31-s − 0.522·33-s − 0.169·35-s + 2/3·36-s + 0.328·37-s + 0.800·39-s − 1.87·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7029112391\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7029112391\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−16.59564054902116243444879277129, −15.27252635693956706959973708757, −14.09621015074921201256906951951, −13.34832607225341458479124517394, −11.80747475770717630386411695404, −10.26175613408033680908306725927, −8.679498853252053444462430241617, −7.980947296320422939728903060714, −5.47907564040983295563221596641, −3.61689003045514298236484607491,
3.61689003045514298236484607491, 5.47907564040983295563221596641, 7.980947296320422939728903060714, 8.679498853252053444462430241617, 10.26175613408033680908306725927, 11.80747475770717630386411695404, 13.34832607225341458479124517394, 14.09621015074921201256906951951, 15.27252635693956706959973708757, 16.59564054902116243444879277129