L(s) = 1 | + 2.42·2-s + 3-s + 3.86·4-s + 5-s + 2.42·6-s − 7-s + 4.50·8-s + 9-s + 2.42·10-s + 2.79·11-s + 3.86·12-s − 5.96·13-s − 2.42·14-s + 15-s + 3.18·16-s + 4.49·17-s + 2.42·18-s + 8.29·19-s + 3.86·20-s − 21-s + 6.76·22-s + 23-s + 4.50·24-s + 25-s − 14.4·26-s + 27-s − 3.86·28-s + ⋯ |
L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.93·4-s + 0.447·5-s + 0.988·6-s − 0.377·7-s + 1.59·8-s + 0.333·9-s + 0.765·10-s + 0.842·11-s + 1.11·12-s − 1.65·13-s − 0.646·14-s + 0.258·15-s + 0.795·16-s + 1.09·17-s + 0.570·18-s + 1.90·19-s + 0.863·20-s − 0.218·21-s + 1.44·22-s + 0.208·23-s + 0.919·24-s + 0.200·25-s − 2.83·26-s + 0.192·27-s − 0.729·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.655444875\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.655444875\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 + 5.96T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 - 8.29T + 19T^{2} \) |
| 29 | \( 1 + 0.288T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 8.10T + 41T^{2} \) |
| 43 | \( 1 + 0.264T + 43T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 4.95T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 + 8.27T + 97T^{2} \) |
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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
−9.113969452578834847651896205547, −7.891197094279031102259257274422, −7.05226656463285299835135525097, −6.65680452194706990355515025696, −5.38180290092489709168524328359, −5.19194447539232595761022585808, −4.05723701831019681276497910727, −3.22641134148286708355922430123, −2.68701059242915819186467737345, −1.48611365306360526968184044986,
1.48611365306360526968184044986, 2.68701059242915819186467737345, 3.22641134148286708355922430123, 4.05723701831019681276497910727, 5.19194447539232595761022585808, 5.38180290092489709168524328359, 6.65680452194706990355515025696, 7.05226656463285299835135525097, 7.891197094279031102259257274422, 9.113969452578834847651896205547