L(s) = 1 | + (0.592 − 0.342i)7-s + (0.326 + 0.118i)13-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)25-s + (1.11 − 0.642i)31-s + 1.53·37-s + (−1.26 − 0.223i)43-s + (−0.266 + 0.460i)49-s + (0.0603 + 0.342i)61-s + (−0.439 − 0.524i)67-s + (0.326 − 0.118i)73-s + (−0.233 − 0.642i)79-s + (0.233 − 0.0412i)91-s + (0.766 + 0.642i)97-s + (1.70 + 0.984i)103-s + ⋯ |
L(s) = 1 | + (0.592 − 0.342i)7-s + (0.326 + 0.118i)13-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)25-s + (1.11 − 0.642i)31-s + 1.53·37-s + (−1.26 − 0.223i)43-s + (−0.266 + 0.460i)49-s + (0.0603 + 0.342i)61-s + (−0.439 − 0.524i)67-s + (0.326 − 0.118i)73-s + (−0.233 − 0.642i)79-s + (0.233 − 0.0412i)91-s + (0.766 + 0.642i)97-s + (1.70 + 0.984i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.341341640\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.341341640\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
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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
−8.897374955836976761068785727133, −8.223370616572615126552229252497, −7.50839631355904625549751011799, −6.67739819043847826785831170394, −5.99318718608738706648989916065, −4.86403843603672319696373064768, −4.41526488262755357749970646050, −3.28744066663394950134247637470, −2.29168992537748781087707492552, −1.05638252967925549042839901036,
1.27312086160543450059328495415, 2.38762544390805577242169113432, 3.38042380681135728129232329399, 4.42394146450408279512184821282, 5.09382660699186942165027402880, 6.05430973402807058567681790956, 6.65281001037507174989899480654, 7.69741966421245883611813088352, 8.352301395626404260493024462741, 8.834131759589735280823414992410