L(s) = 1 | + 3-s − 2·4-s + 5-s − 7-s − 2·9-s + 3·11-s − 2·12-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 + 12·41-s − 10·43-s − 6·44-s − 2·45-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 + 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 + 1.87·41-s − 1.52·43-s − 0.904·44-s − 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.754572510\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.754572510\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.378748037315465445192790829210, −7.58370080180617520189810021223, −6.52421319630927198344725128175, −5.95302503292927073396566116958, −5.25579521602379928421249093772, −4.29829876881027420494319425466, −3.66964895044272310843171490096, −2.92042524544419550067578696465, −1.88740825483186720696747488093, −0.68446346217697022883479913777,
0.68446346217697022883479913777, 1.88740825483186720696747488093, 2.92042524544419550067578696465, 3.66964895044272310843171490096, 4.29829876881027420494319425466, 5.25579521602379928421249093772, 5.95302503292927073396566116958, 6.52421319630927198344725128175, 7.58370080180617520189810021223, 8.378748037315465445192790829210