| L(s) = 1 | − 2·3-s + 3·7-s + 9-s + 5·11-s + 5·13-s + 4·17-s + 19-s − 6·21-s + 23-s + 4·27-s + 9·29-s − 2·31-s − 10·33-s + 2·37-s − 10·39-s + 3·41-s + 7·43-s + 12·47-s + 2·49-s − 8·51-s − 12·53-s − 2·57-s − 6·59-s − 10·61-s + 3·63-s + 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 1.38·13-s + 0.970·17-s + 0.229·19-s − 1.30·21-s + 0.208·23-s + 0.769·27-s + 1.67·29-s − 0.359·31-s − 1.74·33-s + 0.328·37-s − 1.60·39-s + 0.468·41-s + 1.06·43-s + 1.75·47-s + 2/7·49-s − 1.12·51-s − 1.64·53-s − 0.264·57-s − 0.781·59-s − 1.28·61-s + 0.377·63-s + 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990129452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990129452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.350321369612509992874025462338, −7.56435142562007891775260409519, −6.65297251918000349456289919038, −6.05958714158083343162708402996, −5.53580624472769633330997403662, −4.61212767822177650453600644877, −4.04209567534456845219323238659, −2.97072320071717046138772470200, −1.40081065739806276247046171389, −1.01560926897813459933367309440,
1.01560926897813459933367309440, 1.40081065739806276247046171389, 2.97072320071717046138772470200, 4.04209567534456845219323238659, 4.61212767822177650453600644877, 5.53580624472769633330997403662, 6.05958714158083343162708402996, 6.65297251918000349456289919038, 7.56435142562007891775260409519, 8.350321369612509992874025462338
