L(s) = 1 | − 0.519·3-s − 4.76·7-s − 2.72·9-s − 0.960·11-s + 2.24·13-s − 0.249·17-s − 19-s + 2.48·21-s + 9.01·23-s + 2.97·27-s + 6.24·29-s − 2.96·31-s + 0.499·33-s − 0.0399·37-s − 1.16·39-s + 4.96·43-s − 9.49·47-s + 15.7·49-s + 0.129·51-s − 6.84·53-s + 0.519·57-s + 14.5·59-s + 7.53·61-s + 13.0·63-s − 5.72·67-s − 4.68·69-s + 9.61·71-s + ⋯ |
L(s) = 1 | − 0.300·3-s − 1.80·7-s − 0.909·9-s − 0.289·11-s + 0.623·13-s − 0.0605·17-s − 0.229·19-s + 0.541·21-s + 1.88·23-s + 0.573·27-s + 1.16·29-s − 0.531·31-s + 0.0868·33-s − 0.00656·37-s − 0.187·39-s + 0.756·43-s − 1.38·47-s + 2.24·49-s + 0.0181·51-s − 0.940·53-s + 0.0688·57-s + 1.89·59-s + 0.965·61-s + 1.64·63-s − 0.699·67-s − 0.564·69-s + 1.14·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 0.519T + 3T^{2} \) |
| 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 + 0.960T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 0.249T + 17T^{2} \) |
| 23 | \( 1 - 9.01T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 + 0.0399T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 + 5.72T + 67T^{2} \) |
| 71 | \( 1 - 9.61T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 6.07T + 79T^{2} \) |
| 83 | \( 1 + 7.45T + 83T^{2} \) |
| 89 | \( 1 - 4.07T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.30984838344436363852148403113, −6.71799273994787651919360459405, −6.20441526936187845998956384054, −5.56629691080129527307689124835, −4.80981871729465203824659990143, −3.73156856980757294314803565768, −3.08244234188786161215824687990, −2.56321340569723069666411880923, −0.998660900498124719049189214261, 0,
0.998660900498124719049189214261, 2.56321340569723069666411880923, 3.08244234188786161215824687990, 3.73156856980757294314803565768, 4.80981871729465203824659990143, 5.56629691080129527307689124835, 6.20441526936187845998956384054, 6.71799273994787651919360459405, 7.30984838344436363852148403113