| L(s) = 1 | + 2.19·2-s − 2.48·3-s + 2.81·4-s − 5.46·6-s + 3.05·7-s + 1.78·8-s + 3.19·9-s + 5.36·11-s − 7.00·12-s + 4.59·13-s + 6.70·14-s − 1.70·16-s + 7.01·18-s + 4.57·19-s − 7.60·21-s + 11.7·22-s + 1.24·23-s − 4.45·24-s + 10.0·26-s − 0.483·27-s + 8.60·28-s + 5.93·29-s − 9.84·31-s − 7.31·32-s − 13.3·33-s + 8.99·36-s + 4.20·37-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 1.43·3-s + 1.40·4-s − 2.22·6-s + 1.15·7-s + 0.632·8-s + 1.06·9-s + 1.61·11-s − 2.02·12-s + 1.27·13-s + 1.79·14-s − 0.425·16-s + 1.65·18-s + 1.04·19-s − 1.66·21-s + 2.50·22-s + 0.260·23-s − 0.909·24-s + 1.97·26-s − 0.0931·27-s + 1.62·28-s + 1.10·29-s − 1.76·31-s − 1.29·32-s − 2.32·33-s + 1.49·36-s + 0.691·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.388692601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.388692601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 9.84T + 31T^{2} \) |
| 37 | \( 1 - 4.20T + 37T^{2} \) |
| 41 | \( 1 + 0.404T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 4.97T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 2.27T + 97T^{2} \) |
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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
−7.48763997053348117845573848814, −6.84837730256476577200087734569, −6.17855482539848851682538285035, −5.72399987807834525707763973087, −5.15817649141953809986908082615, −4.39416368260626326446115874966, −3.97884878694900529045084180351, −3.06707426172861571797765543184, −1.66487484101355480102246500765, −1.01177707025130937648839010207,
1.01177707025130937648839010207, 1.66487484101355480102246500765, 3.06707426172861571797765543184, 3.97884878694900529045084180351, 4.39416368260626326446115874966, 5.15817649141953809986908082615, 5.72399987807834525707763973087, 6.17855482539848851682538285035, 6.84837730256476577200087734569, 7.48763997053348117845573848814
