| L(s) = 1 | − 3-s + 7-s + 9-s + 4·11-s + 3·13-s − 4·17-s + 19-s − 21-s − 27-s + 8·29-s + 31-s − 4·33-s − 2·37-s − 3·39-s + 2·41-s + 11·43-s − 2·47-s − 6·49-s + 4·51-s + 10·53-s − 57-s + 6·59-s − 11·61-s + 63-s − 9·67-s + 6·71-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.832·13-s − 0.970·17-s + 0.229·19-s − 0.218·21-s − 0.192·27-s + 1.48·29-s + 0.179·31-s − 0.696·33-s − 0.328·37-s − 0.480·39-s + 0.312·41-s + 1.67·43-s − 0.291·47-s − 6/7·49-s + 0.560·51-s + 1.37·53-s − 0.132·57-s + 0.781·59-s − 1.40·61-s + 0.125·63-s − 1.09·67-s + 0.712·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911475250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911475250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.384131869259293106252052362931, −7.46967197478612318229888839436, −6.64055326081393237561653594287, −6.25185694051602566924925819670, −5.40179159821898051000768876089, −4.45251719385586221727772231176, −4.01854876380914976774698358195, −2.88633184494197132886473702903, −1.70066953601413916053657398768, −0.838705118468641754740772685119,
0.838705118468641754740772685119, 1.70066953601413916053657398768, 2.88633184494197132886473702903, 4.01854876380914976774698358195, 4.45251719385586221727772231176, 5.40179159821898051000768876089, 6.25185694051602566924925819670, 6.64055326081393237561653594287, 7.46967197478612318229888839436, 8.384131869259293106252052362931
